GPT-4o on 4.OA.B.4
Common Core Standard 4.OA.B.4
This standard covers the following:
Find all factor pairs for a whole number in the range 1-100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1-100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1-100 is prime or composite.
Problem: Find all factor pairs of 64.
GPT-4o:
To find all factor pairs of 64, we need to identify all pairs of integers that multiply together to give 64. 1. Start with 1: \[ 1 \times 64 = 64 \] So, (1, 64) is a factor pair. 2. Next, try 2: \[ 2 \times 32 = 64 \] So, (2, 32) is a factor pair. 3. Next, try 4: \[ 4 \times 16 = 64 \] So, (4, 16) is a factor pair. 4. Next, try 8: \[ 8 \times 8 = 64 \] So, (8, 8) is a factor pair. Since 8 is the square root of 64, we have found all unique factor pairs. Listing them, we have: - (1, 64) - (2, 32) - (4, 16) - (8, 8) ### Answer: (1, 64), (2, 32), (4, 16), (8, 8)
Expected answer: [(1, 64), (2, 32), (4, 16), (8, 8)]
Correct? Yes
Problem: List all the factor pairs of 18.
GPT-4o:
To find all the factor pairs of 18, we need to identify all pairs of integers that multiply together to give 18. 1. Start with 1: - \(1 \times 18 = 18\) - Factor pair: (1, 18) 2. Next, try 2: - \(2 \times 9 = 18\) - Factor pair: (2, 9) 3. Next, try 3: - \(3 \times 6 = 18\) - Factor pair: (3, 6) 4. Next, try 4: - \(4 \times 4.5 = 18\) (4.5 is not an integer, so this is not a valid factor pair) 5. Next, try 5: - \(5 \times 3.6 = 18\) (3.6 is not an integer, so this is not a valid factor pair) 6. Finally, try 6: - \(6 \times 3 = 18\) (This is the same as the pair (3, 6), which we already have) Since we have checked all integers up to the square root of 18, we have found all the factor pairs. ### Answer: (1, 18), (2, 9), (3, 6)
Expected answer: [(1, 18), (2, 9), (3, 6)]
Correct? Yes
Problem: Find all factor pairs of 28.
GPT-4o:
To find all factor pairs of 28, we need to determine which pairs of integers multiply together to give the product of 28.
1. Start with 1:
\[
1 \times 28 = 28
\]
So, (1, 28) is a factor pair.
2. Next, try 2:
\[
2 \times 14 = 28
\]
So, (2, 14) is a factor pair.
3. Next, try 3:
\[
3 \times \text{(no integer)} = 28
\]
Since 28 is not divisible by 3, there is no factor pair involving 3.
4. Next, try 4:
\[
4 \times 7 = 28
\]
So, (4, 7) is a factor pair.
5. Next, try 5:
\[
5 \times \text{(no integer)} = 28
\]
Since 28 is not divisible by 5, there is no factor pair involving 5.
6. Next, try 6:
\[
6 \times \text{(no integer)} = 28
\]
Since 28 is not divisible by 6, there is no factor pair involving 6.
Since we have checked all integers up to the square root of 28 (approximately 5.29), we have found all factor pairs.
### Answer: (1, 28), (2, 14), (4, 7)
Expected answer: [(1, 28), (2, 14), (4, 7)]
Correct? Yes
Problem: List all factor pairs of the number 48.
GPT-4o:
To find all factor pairs of the number 48, we need to identify all pairs of integers that multiply together to give 48. 1. Start with 1: - \(1 \times 48 = 48\) - Factor pair: (1, 48) 2. Next, try 2: - \(2 \times 24 = 48\) - Factor pair: (2, 24) 3. Next, try 3: - \(3 \times 16 = 48\) - Factor pair: (3, 16) 4. Next, try 4: - \(4 \times 12 = 48\) - Factor pair: (4, 12) 5. Next, try 6: - \(6 \times 8 = 48\) - Factor pair: (6, 8) 6. Next, try 8: - \(8 \times 6 = 48\) - This pair is already listed as (6, 8), so we stop here. Thus, the factor pairs of 48 are: (1, 48), (2, 24), (3, 16), (4, 12), and (6, 8). ### Answer: (1, 48), (2, 24), (3, 16), (4, 12), (6, 8)
Expected answer: [(1, 48), (2, 24), (3, 16), (4, 12), (6, 8)]
Correct? Yes
Problem: Write down all factor pairs of 7.
GPT-4o:
To find the factor pairs of 7, we need to identify all pairs of integers that multiply together to give the product of 7. 1. Start with 1: - \(1 \times 7 = 7\) Since 7 is a prime number, it has no other factors besides 1 and itself. Thus, the factor pairs of 7 are: - (1, 7) - (7, 1) ### Answer: (1, 7) and (7, 1)
Expected answer: [(1, 7)]
Correct? Yes
Problem: Write down every pair of factors of 98.
GPT-4o:
To find the pairs of factors of 98, we need to identify all the numbers that multiply together to give 98.
1. Start with 1:
- \(1 \times 98 = 98\)
- Pair: (1, 98)
2. Next, check 2:
- \(2 \times 49 = 98\)
- Pair: (2, 49)
3. Check 3:
- 98 is not divisible by 3 (since \(98 \div 3 \approx 32.67\))
4. Check 4:
- 98 is not divisible by 4 (since \(98 \div 4 = 24.5\))
5. Check 5:
- 98 is not divisible by 5 (since it does not end in 0 or 5)
6. Check 6:
- 98 is not divisible by 6 (since \(98 \div 6 \approx 16.33\))
7. Check 7:
- \(7 \times 14 = 98\)
- Pair: (7, 14)
8. Check 8:
- 98 is not divisible by 8 (since \(98 \div 8 \approx 12.25\))
9. Check 9:
- 98 is not divisible by 9 (since \(98 \div 9 \approx 10.89\))
10. Check 10:
- 98 is not divisible by 10 (since it does not end in 0)
11. Check 11:
- 98 is not divisible by 11 (since \(98 \div 11 \approx 8.91\))
12. Check 12:
- 98 is not divisible by 12 (since \(98 \div 12 \approx 8.17\))
13. Check 13:
- 98 is not divisible by 13 (since \(98 \div 13 \approx 7.54\))
14. Check 14:
- \(14 \times 7 = 98\) (already listed as (7, 14))
15. Check 49:
- \(49 \times 2 = 98\) (already listed as (2, 49))
16. Check 98:
- \(98 \times 1 = 98\) (already listed as (1, 98))
Thus, the pairs of factors of 98 are:
(1, 98), (2, 49), and (7, 14).
### Answer: (1, 98), (2, 49), (7, 14)
Expected answer: [(1, 98), (2, 49), (7, 14)]
Correct? Yes
Problem: List all the factor pairs of the number 26.
GPT-4o:
To find the factor pairs of the number 26, we need to identify all pairs of integers that multiply together to give 26.
1. Start with 1:
- \(1 \times 26 = 26\)
- So, (1, 26) is a factor pair.
2. Next, try 2:
- \(2 \times 13 = 26\)
- So, (2, 13) is a factor pair.
3. Check 3:
- \(3 \times \text{(not an integer)} = 26\)
- So, 3 is not a factor.
4. Check 4:
- \(4 \times \text{(not an integer)} = 26\)
- So, 4 is not a factor.
5. Check 5:
- \(5 \times \text{(not an integer)} = 26\)
- So, 5 is not a factor.
Since 6 is greater than the square root of 26 (approximately 5.1), we can stop here.
Thus, the factor pairs of 26 are:
- (1, 26)
- (2, 13)
### Answer: (1, 26) and (2, 13)
Expected answer: [(1, 26), (2, 13)]
Correct? Yes
Problem: Write down all factor pairs of 59.
GPT-4o:
To find the factor pairs of 59, we need to determine which pairs of integers multiply together to give 59. First, we check if 59 is a prime number. A prime number has no divisors other than 1 and itself. We test divisibility by all prime numbers less than the square root of 59 (approximately 7.68). - 59 is not divisible by 2 (it's odd). - 59 is not divisible by 3 (sum of digits, 5 + 9 = 14, is not divisible by 3). - 59 is not divisible by 5 (it does not end in 0 or 5). - 59 is not divisible by 7 (59 ÷ 7 ≈ 8.4286, not an integer). Since 59 is not divisible by any of these primes, it is a prime number. Therefore, the only factor pairs of 59 are (1, 59). ### Answer: (1, 59)
Expected answer: [(1, 59)]
Correct? Yes
Problem: Find out all factor pairs of 20.
GPT-4o:
To find all factor pairs of 20, we need to determine which pairs of integers multiply together to give the product of 20.
1. Start with 1:
\( 1 \times 20 = 20 \)
So, (1, 20) is a factor pair.
2. Next, try 2:
\( 2 \times 10 = 20 \)
So, (2, 10) is a factor pair.
3. Next, try 4:
\( 4 \times 5 = 20 \)
So, (4, 5) is a factor pair.
4. Check 3:
\( 3 \times \text{(not an integer)} = 20 \)
So, 3 is not a factor.
Since we have checked all integers up to the square root of 20 (approximately 4.47), we have found all factor pairs.
### Answer: (1, 20), (2, 10), (4, 5)
Expected answer: [(1, 20), (2, 10), (4, 5)]
Correct? Yes
Problem: Write down all factor pairs of 20.
GPT-4o:
To find all factor pairs of 20, we need to identify all pairs of integers that multiply together to give 20.
1. Start with 1:
- \(1 \times 20 = 20\)
- Factor pair: (1, 20)
2. Next, try 2:
- \(2 \times 10 = 20\)
- Factor pair: (2, 10)
3. Next, try 4:
- \(4 \times 5 = 20\)
- Factor pair: (4, 5)
4. Check 3:
- \(3 \times \text{(no integer)} = 20\)
- No factor pair here.
5. Continue checking higher numbers until the square root of 20 (approximately 4.47):
- We have already checked 1, 2, and 4.
Thus, the factor pairs of 20 are:
- (1, 20)
- (2, 10)
- (4, 5)
### Answer: (1, 20), (2, 10), (4, 5)
Expected answer: [(1, 20), (2, 10), (4, 5)]
Correct? Yes
Problem: Please list down all factor pairs of 13.
GPT-4o:
To find the factor pairs of 13, we need to identify all pairs of integers that multiply together to give 13. 1. Start with 1: - \(1 \times 13 = 13\) Since 13 is a prime number, it has no other factors besides 1 and itself. Thus, the factor pairs of 13 are: - (1, 13) ### Answer: (1, 13)
Expected answer: [(1, 13)]
Correct? Yes
Problem: Find all factor pairs of 81.
GPT-4o:
To find all factor pairs of 81, we need to identify all pairs of integers that multiply together to give 81.
1. Start with 1:
\[
1 \times 81 = 81
\]
So, (1, 81) is a factor pair.
2. Next, try 2:
\[
2 \times \text{(no integer)} = 81
\]
Since 81 is not divisible by 2, 2 is not a factor.
3. Try 3:
\[
3 \times 27 = 81
\]
So, (3, 27) is a factor pair.
4. Try 4:
\[
4 \times \text{(no integer)} = 81
\]
Since 81 is not divisible by 4, 4 is not a factor.
5. Try 5:
\[
5 \times \text{(no integer)} = 81
\]
Since 81 is not divisible by 5, 5 is not a factor.
6. Try 6:
\[
6 \times \text{(no integer)} = 81
\]
Since 81 is not divisible by 6, 6 is not a factor.
7. Try 7:
\[
7 \times \text{(no integer)} = 81
\]
Since 81 is not divisible by 7, 7 is not a factor.
8. Try 8:
\[
8 \times \text{(no integer)} = 81
\]
Since 81 is not divisible by 8, 8 is not a factor.
9. Try 9:
\[
9 \times 9 = 81
\]
So, (9, 9) is a factor pair.
Since we have tried all integers up to the square root of 81 (which is 9), we have found all factor pairs.
### Answer: (1, 81), (3, 27), (9, 9)
Expected answer: [(1, 81), (3, 27), (9, 9)]
Correct? Yes
Problem: List all factor pairs of 76.
GPT-4o:
To find all factor pairs of 76, we need to identify all pairs of integers that multiply together to give 76.
1. Start with 1:
\( 1 \times 76 = 76 \)
So, (1, 76) is a factor pair.
2. Next, try 2:
\( 2 \times 38 = 76 \)
So, (2, 38) is a factor pair.
3. Try 3:
\( 3 \times \text{(not an integer)} \neq 76 \)
So, 3 is not a factor.
4. Try 4:
\( 4 \times 19 = 76 \)
So, (4, 19) is a factor pair.
5. Try 5:
\( 5 \times \text{(not an integer)} \neq 76 \)
So, 5 is not a factor.
6. Try 6:
\( 6 \times \text{(not an integer)} \neq 76 \)
So, 6 is not a factor.
7. Try 7:
\( 7 \times \text{(not an integer)} \neq 76 \)
So, 7 is not a factor.
8. Try 8:
\( 8 \times \text{(not an integer)} \neq 76 \)
So, 8 is not a factor.
Since we have checked up to the square root of 76 (approximately 8.7), we have found all factor pairs.
### Answer: (1, 76), (2, 38), (4, 19)
Expected answer: [(1, 76), (2, 38), (4, 19)]
Correct? Yes
Problem: Write down all factor pairs of 50.
GPT-4o:
To find all factor pairs of 50, we need to identify all pairs of integers that multiply together to give 50. 1. Start with 1: - \(1 \times 50 = 50\) - Factor pair: (1, 50) 2. Next, try 2: - \(2 \times 25 = 50\) - Factor pair: (2, 25) 3. Next, try 3: - 50 is not divisible by 3, so no factor pair here. 4. Next, try 4: - 50 is not divisible by 4, so no factor pair here. 5. Next, try 5: - \(5 \times 10 = 50\) - Factor pair: (5, 10) 6. Next, try 6: - 50 is not divisible by 6, so no factor pair here. 7. Continue this process until the square root of 50 (approximately 7.07): - 7 is not a factor of 50. Thus, the factor pairs of 50 are: - (1, 50) - (2, 25) - (5, 10) ### Answer: (1, 50), (2, 25), (5, 10)
Expected answer: [(1, 50), (2, 25), (5, 10)]
Correct? Yes
Problem: List all the factor pairs of the number 76.
GPT-4o:
To find the factor pairs of 76, we need to identify all pairs of integers that multiply together to give 76.
1. Start with 1:
\( 1 \times 76 = 76 \)
So, (1, 76) is a factor pair.
2. Next, try 2:
\( 2 \times 38 = 76 \)
So, (2, 38) is a factor pair.
3. Next, try 3:
\( 3 \times \text{(not an integer)} = 76 \)
So, 3 is not a factor.
4. Next, try 4:
\( 4 \times 19 = 76 \)
So, (4, 19) is a factor pair.
5. Next, try 5:
\( 5 \times \text{(not an integer)} = 76 \)
So, 5 is not a factor.
6. Next, try 6:
\( 6 \times \text{(not an integer)} = 76 \)
So, 6 is not a factor.
7. Next, try 7:
\( 7 \times \text{(not an integer)} = 76 \)
So, 7 is not a factor.
8. Next, try 8:
\( 8 \times \text{(not an integer)} = 76 \)
So, 8 is not a factor.
Since we have checked all integers up to the square root of 76 (approximately 8.7), we have found all factor pairs.
### Answer: (1, 76), (2, 38), (4, 19)
Expected answer: [(1, 76), (2, 38), (4, 19)]
Correct? Yes
Problem: List all factor pairs of 56.
GPT-4o:
To find all factor pairs of 56, we need to identify all pairs of integers that multiply together to give 56.
1. Start with 1:
\( 1 \times 56 = 56 \)
So, (1, 56) is a factor pair.
2. Next, try 2:
\( 2 \times 28 = 56 \)
So, (2, 28) is a factor pair.
3. Next, try 3:
\( 3 \times \text{(not an integer)} = 56 \)
So, 3 is not a factor.
4. Next, try 4:
\( 4 \times 14 = 56 \)
So, (4, 14) is a factor pair.
5. Next, try 5:
\( 5 \times \text{(not an integer)} = 56 \)
So, 5 is not a factor.
6. Next, try 6:
\( 6 \times \text{(not an integer)} = 56 \)
So, 6 is not a factor.
7. Next, try 7:
\( 7 \times 8 = 56 \)
So, (7, 8) is a factor pair.
Since 7 and 8 are close to each other, we have found all factor pairs.
### Answer: (1, 56), (2, 28), (4, 14), (7, 8)
Expected answer: [(1, 56), (2, 28), (4, 14), (7, 8)]
Correct? Yes
Problem: List all the factor pairs of the number 44.
GPT-4o:
To find the factor pairs of 44, we need to identify all pairs of integers that multiply together to give 44.
1. Start with 1:
\[
1 \times 44 = 44
\]
So, (1, 44) is a factor pair.
2. Next, try 2:
\[
2 \times 22 = 44
\]
So, (2, 22) is a factor pair.
3. Next, try 3:
\[
3 \times \text{(not an integer)} = 44
\]
So, 3 is not a factor.
4. Next, try 4:
\[
4 \times 11 = 44
\]
So, (4, 11) is a factor pair.
5. Next, try 5:
\[
5 \times \text{(not an integer)} = 44
\]
So, 5 is not a factor.
6. Next, try 6:
\[
6 \times \text{(not an integer)} = 44
\]
So, 6 is not a factor.
7. Next, try 7:
\[
7 \times \text{(not an integer)} = 44
\]
So, 7 is not a factor.
8. Next, try 8:
\[
8 \times \text{(not an integer)} = 44
\]
So, 8 is not a factor.
9. Next, try 9:
\[
9 \times \text{(not an integer)} = 44
\]
So, 9 is not a factor.
10. Finally, try 10:
\[
10 \times \text{(not an integer)} = 44
\]
So, 10 is not a factor.
Since we have tried all integers up to the square root of 44, we have found all factor pairs.
### Answer: (1, 44), (2, 22), (4, 11)
Expected answer: [(1, 44), (2, 22), (4, 11)]
Correct? Yes
Problem: List all the factor pairs of 5.
GPT-4o:
To find the factor pairs of 5, we need to identify all pairs of integers that multiply together to give the product of 5. 1. Start with 1: - \(1 \times 5 = 5\) Since 5 is a prime number, it has no other factors besides 1 and itself. Thus, the factor pairs of 5 are: - (1, 5) - (5, 1) ### Answer: (1, 5) and (5, 1)
Expected answer: [(1, 5)]
Correct? Yes
Problem: Identify all factor pairs of the number 95.
GPT-4o:
To identify all factor pairs of the number 95, we need to find pairs of integers that multiply together to give 95.
1. Start with 1:
\[
1 \times 95 = 95
\]
So, (1, 95) is a factor pair.
2. Check the next integer, 2:
\[
95 \div 2 = 47.5
\]
Since 47.5 is not an integer, 2 is not a factor.
3. Check the next integer, 3:
\[
95 \div 3 \approx 31.67
\]
Since 31.67 is not an integer, 3 is not a factor.
4. Check the next integer, 4:
\[
95 \div 4 = 23.75
\]
Since 23.75 is not an integer, 4 is not a factor.
5. Check the next integer, 5:
\[
95 \div 5 = 19
\]
Since 19 is an integer, (5, 19) is a factor pair.
6. Check the next integer, 6:
\[
95 \div 6 \approx 15.83
\]
Since 15.83 is not an integer, 6 is not a factor.
7. Continue this process up to the square root of 95 (approximately 9.75). Since we have already checked all integers up to 9, we can stop here.
The factor pairs of 95 are:
\[
(1, 95) \quad \text{and} \quad (5, 19)
\]
### Answer: (1, 95) and (5, 19)
Expected answer: [(1, 95), (5, 19)]
Correct? Yes
Problem: Please list all the factor pairs of 36.
GPT-4o:
To find all the factor pairs of 36, we need to identify all pairs of integers that multiply together to give 36. 1. Start with 1: - \(1 \times 36 = 36\) - Factor pair: (1, 36) 2. Next, try 2: - \(2 \times 18 = 36\) - Factor pair: (2, 18) 3. Next, try 3: - \(3 \times 12 = 36\) - Factor pair: (3, 12) 4. Next, try 4: - \(4 \times 9 = 36\) - Factor pair: (4, 9) 5. Next, try 6: - \(6 \times 6 = 36\) - Factor pair: (6, 6) Since 6 is the square root of 36, we have now listed all unique factor pairs. ### Answer: (1, 36), (2, 18), (3, 12), (4, 9), (6, 6)
Expected answer: [(1, 36), (2, 18), (3, 12), (4, 9), (6, 6)]
Correct? Yes
Problem: Compute all the factor pairs of the number 59.
GPT-4o:
To find the factor pairs of the number 59, we need to determine which pairs of integers multiply together to give 59. First, we check if 59 is a prime number. A prime number has no divisors other than 1 and itself. We test divisibility by all prime numbers less than the square root of 59 (approximately 7.68). - 59 is not divisible by 2 (it's odd). - 59 is not divisible by 3 (sum of digits, 5 + 9 = 14, is not divisible by 3). - 59 is not divisible by 5 (it does not end in 0 or 5). - 59 is not divisible by 7 (59 ÷ 7 ≈ 8.4286, not an integer). Since 59 is not divisible by any of these primes, it is a prime number. Therefore, the only factor pairs of 59 are (1, 59) and (59, 1). ### Answer: (1, 59)
Expected answer: [(1, 59)]
Correct? Yes
Problem: What are all the factor pairs of 3?
GPT-4o:
To find the factor pairs of 3, we need to identify all pairs of integers that multiply together to give the product of 3. 1. Start with 1: - \(1 \times 3 = 3\) - This gives us the factor pair (1, 3). 2. Check for any other pairs: - Since 3 is a prime number, it has no other factors besides 1 and itself. Thus, the only factor pairs of 3 are (1, 3) and (3, 1). ### Answer: (1, 3) and (3, 1)
Expected answer: [(1, 3)]
Correct? Yes
Problem: Could you list all factor pairs of 51?
GPT-4o:
To find the factor pairs of 51, we need to identify all pairs of integers that multiply together to give 51. 1. Start with 1: - \(1 \times 51 = 51\) - So, (1, 51) is a factor pair. 2. Check the next integer, 2: - 51 is not divisible by 2 (since 51 is odd). 3. Check the next integer, 3: - \(51 \div 3 = 17\) - So, \(3 \times 17 = 51\) - Therefore, (3, 17) is a factor pair. 4. Check the next integer, 4: - 51 is not divisible by 4. 5. Check the next integer, 5: - 51 is not divisible by 5 (since it does not end in 0 or 5). 6. Check the next integer, 6: - 51 is not divisible by 6. 7. Continue this process up to the square root of 51 (approximately 7.14): - 51 is not divisible by 7. Since we have checked all integers up to the square root of 51, we have found all factor pairs. ### Answer: (1, 51) and (3, 17)
Expected answer: [(1, 51), (3, 17)]
Correct? Yes
Problem: List all factor pairs of 50.
GPT-4o:
To find all factor pairs of 50, we need to identify all pairs of integers that multiply together to give 50.
1. Start with 1:
- \(1 \times 50 = 50\)
- Factor pair: (1, 50)
2. Next, try 2:
- \(2 \times 25 = 50\)
- Factor pair: (2, 25)
3. Next, try 3:
- \(3 \times \text{(not an integer)} = 50\)
- No factor pair
4. Next, try 4:
- \(4 \times \text{(not an integer)} = 50\)
- No factor pair
5. Next, try 5:
- \(5 \times 10 = 50\)
- Factor pair: (5, 10)
6. Next, try 6:
- \(6 \times \text{(not an integer)} = 50\)
- No factor pair
7. Continue this process until the square root of 50 (approximately 7.07):
- No other integers between 1 and 7 multiply to 50.
Thus, the factor pairs of 50 are:
- (1, 50)
- (2, 25)
- (5, 10)
### Answer: (1, 50), (2, 25), (5, 10)
Expected answer: [(1, 50), (2, 25), (5, 10)]
Correct? Yes
Problem: Write down all factor pairs of 47.
GPT-4o:
To find the factor pairs of 47, we need to identify all pairs of integers that multiply to give 47. First, we note that 47 is a prime number, meaning it has no divisors other than 1 and itself. Therefore, the only factor pairs of 47 are: 1. (1, 47) 2. (47, 1) These pairs represent the same factors, just in different orders. ### Answer: (1, 47)
Expected answer: [(1, 47)]
Correct? Yes
Problem: Write down all factor pairs of 51.
GPT-4o:
To find all factor pairs of 51, we need to identify all pairs of integers that multiply together to give 51. 1. Start with 1: - \(1 \times 51 = 51\) - Factor pair: (1, 51) 2. Check the next integer, 2: - 51 is not divisible by 2 (since 51 is odd). 3. Check the next integer, 3: - \(51 \div 3 = 17\) - \(3 \times 17 = 51\) - Factor pair: (3, 17) 4. Check the next integer, 4: - 51 is not divisible by 4. 5. Check the next integer, 5: - 51 is not divisible by 5. 6. Check the next integer, 6: - 51 is not divisible by 6. 7. Continue this process until the square root of 51 (approximately 7.14): - 51 is not divisible by 7. Since we have checked all integers up to the square root of 51, we have found all factor pairs. ### Answer: (1, 51) and (3, 17)
Expected answer: [(1, 51), (3, 17)]
Correct? Yes
Problem: List all factor pairs of 60.
GPT-4o:
To find all factor pairs of 60, we need to identify all pairs of integers that multiply together to give 60. 1. Start with 1: - \(1 \times 60 = 60\) - Factor pair: (1, 60) 2. Next, try 2: - \(2 \times 30 = 60\) - Factor pair: (2, 30) 3. Next, try 3: - \(3 \times 20 = 60\) - Factor pair: (3, 20) 4. Next, try 4: - \(4 \times 15 = 60\) - Factor pair: (4, 15) 5. Next, try 5: - \(5 \times 12 = 60\) - Factor pair: (5, 12) 6. Next, try 6: - \(6 \times 10 = 60\) - Factor pair: (6, 10) 7. Finally, try 7, 8, and 9: - None of these numbers multiply with another integer to give 60. So, the factor pairs of 60 are: (1, 60), (2, 30), (3, 20), (4, 15), (5, 12), and (6, 10). ### Answer: (1, 60), (2, 30), (3, 20), (4, 15), (5, 12), (6, 10)
Expected answer: [(1, 60), (2, 30), (3, 20), (4, 15), (5, 12), (6, 10)]
Correct? Yes
Problem: List all factor pairs of 43.
GPT-4o:
To find the factor pairs of 43, we need to identify all pairs of integers that multiply together to give 43. First, we note that 43 is a prime number, meaning it has no divisors other than 1 and itself. Thus, the only factor pairs of 43 are: 1 and 43. ### Answer: (1, 43)
Expected answer: [(1, 43)]
Correct? Yes
Problem: List all factor pairs of 70.
GPT-4o:
To find all factor pairs of 70, we need to identify all pairs of integers that multiply together to give 70. 1. Start with 1: - \(1 \times 70 = 70\) - Factor pair: (1, 70) 2. Next, try 2: - \(2 \times 35 = 70\) - Factor pair: (2, 35) 3. Next, try 5: - \(5 \times 14 = 70\) - Factor pair: (5, 14) 4. Next, try 7: - \(7 \times 10 = 70\) - Factor pair: (7, 10) Since 8 and higher numbers up to the square root of 70 (approximately 8.37) do not divide 70 evenly, we have found all factor pairs. ### Answer: (1, 70), (2, 35), (5, 14), (7, 10)
Expected answer: [(1, 70), (2, 35), (5, 14), (7, 10)]
Correct? Yes
Problem: Please list all the factors of the number 2.
GPT-4o:
To find the factors of the number 2, we need to determine which whole numbers can be multiplied together to produce 2. 1. The number 1 is a factor of every integer. 2. The number 2 itself is a factor because 2 * 1 = 2. Therefore, the factors of 2 are 1 and 2. ### Answer: 1, 2
Expected answer: [(1, 2)]
Correct? Yes
Problem: List all the factor pairs of 49.
GPT-4o:
To find the factor pairs of 49, we need to identify all pairs of integers that multiply together to give 49.
1. Start with 1:
\( 1 \times 49 = 49 \)
So, (1, 49) is a factor pair.
2. Next, check 2:
\( 2 \times \text{(no integer)} = 49 \)
49 is not divisible by 2, so there is no factor pair involving 2.
3. Check 3:
\( 3 \times \text{(no integer)} = 49 \)
49 is not divisible by 3, so there is no factor pair involving 3.
4. Check 4:
\( 4 \times \text{(no integer)} = 49 \)
49 is not divisible by 4, so there is no factor pair involving 4.
5. Check 5:
\( 5 \times \text{(no integer)} = 49 \)
49 is not divisible by 5, so there is no factor pair involving 5.
6. Check 6:
\( 6 \times \text{(no integer)} = 49 \)
49 is not divisible by 6, so there is no factor pair involving 6.
7. Check 7:
\( 7 \times 7 = 49 \)
So, (7, 7) is a factor pair.
Since 7 is the square root of 49, we have found all the factor pairs.
### Answer: (1, 49) and (7, 7)
Expected answer: [(1, 49), (7, 7)]
Correct? Yes
Problem: Identify all factor pairs of 17.
GPT-4o:
To identify all factor pairs of 17, we need to find pairs of integers that multiply together to give 17. 1. Start with 1, since 1 is a factor of every integer: \( 1 \times 17 = 17 \) 2. Check if there are any other pairs: - 2 is not a factor of 17 because 17 is not divisible by 2. - 3 is not a factor of 17 because 17 is not divisible by 3. - Continue this process up to the square root of 17 (approximately 4.1). Since 17 is a prime number, it has no other factors besides 1 and itself. Thus, the only factor pair of 17 is: \( (1, 17) \) ### Answer: (1, 17)
Expected answer: [(1, 17)]
Correct? Yes
Problem: Find all the factor pairs of 35.
GPT-4o:
To find all the factor pairs of 35, we need to identify all pairs of integers that multiply together to give 35.
1. Start with 1:
- \(1 \times 35 = 35\)
- So, (1, 35) is a factor pair.
2. Next, try 2:
- \(2 \times \text{(no integer)} = 35\)
- 2 is not a factor of 35.
3. Try 3:
- \(3 \times \text{(no integer)} = 35\)
- 3 is not a factor of 35.
4. Try 4:
- \(4 \times \text{(no integer)} = 35\)
- 4 is not a factor of 35.
5. Try 5:
- \(5 \times 7 = 35\)
- So, (5, 7) is a factor pair.
6. Try 6:
- \(6 \times \text{(no integer)} = 35\)
- 6 is not a factor of 35.
Since we have checked all integers up to the square root of 35 (approximately 5.92), we have found all factor pairs.
### Answer: (1, 35) and (5, 7)
Expected answer: [(1, 35), (5, 7)]
Correct? Yes
Problem: Identify all factor pairs of the number 91.
GPT-4o:
To identify all factor pairs of the number 91, we need to find pairs of integers that multiply together to give 91. 1. Start with 1: \[ 1 \times 91 = 91 \] So, (1, 91) is a factor pair. 2. Check the next integer, 2: \[ 91 \div 2 = 45.5 \] Since 45.5 is not an integer, 2 is not a factor. 3. Check the next integer, 3: \[ 91 \div 3 \approx 30.33 \] Since 30.33 is not an integer, 3 is not a factor. 4. Check the next integer, 4: \[ 91 \div 4 = 22.75 \] Since 22.75 is not an integer, 4 is not a factor. 5. Check the next integer, 5: \[ 91 \div 5 = 18.2 \] Since 18.2 is not an integer, 5 is not a factor. 6. Check the next integer, 6: \[ 91 \div 6 \approx 15.17 \] Since 15.17 is not an integer, 6 is not a factor. 7. Check the next integer, 7: \[ 91 \div 7 = 13 \] Since 13 is an integer, (7, 13) is a factor pair. 8. Check the next integer, 8: \[ 91 \div 8 \approx 11.375 \] Since 11.375 is not an integer, 8 is not a factor. 9. Check the next integer, 9: \[ 91 \div 9 \approx 10.11 \] Since 10.11 is not an integer, 9 is not a factor. Since we have checked all integers up to the square root of 91 (approximately 9.5), we have found all factor pairs. ### Answer: (1, 91) and (7, 13)
Expected answer: [(1, 91), (7, 13)]
Correct? Yes
Problem: List all factor pairs of the number 97.
GPT-4o:
To find the factor pairs of the number 97, we need to determine which pairs of integers multiply together to give 97. First, we check if 97 is a prime number. A prime number has no divisors other than 1 and itself. We test divisibility by all prime numbers less than the square root of 97 (approximately 9.8). - 97 is not divisible by 2 (it's odd). - 97 is not divisible by 3 (sum of digits, 9 + 7 = 16, which is not divisible by 3). - 97 is not divisible by 5 (it does not end in 0 or 5). - 97 is not divisible by 7 (97 ÷ 7 ≈ 13.857, not an integer). Since 97 is not divisible by any prime numbers less than its square root, it is a prime number. Therefore, the only factor pairs of 97 are (1, 97). ### Answer: (1, 97)
Expected answer: [(1, 97)]
Correct? Yes
Problem: What are all the factor pairs of 16?
GPT-4o:
To find all the factor pairs of 16, we need to identify all pairs of integers that multiply together to give 16.
1. Start with 1:
- \(1 \times 16 = 16\)
- Factor pair: (1, 16)
2. Next, try 2:
- \(2 \times 8 = 16\)
- Factor pair: (2, 8)
3. Next, try 3:
- \(3 \times \text{(no integer)} = 16\)
- No factor pair here.
4. Next, try 4:
- \(4 \times 4 = 16\)
- Factor pair: (4, 4)
Since 4 is the square root of 16, we have found all unique factor pairs.
Thus, the factor pairs of 16 are:
- (1, 16)
- (2, 8)
- (4, 4)
### Answer: (1, 16), (2, 8), (4, 4)
Expected answer: [(1, 16), (2, 8), (4, 4)]
Correct? Yes
Problem: Find all the factor pairs of the number 21.
GPT-4o:
To find all the factor pairs of the number 21, we need to identify all pairs of integers that multiply together to give 21.
1. Start with 1:
- \( 1 \times 21 = 21 \)
- So, (1, 21) is a factor pair.
2. Next, try 2:
- \( 2 \times \text{(no integer)} = 21 \)
- 2 is not a factor of 21.
3. Try 3:
- \( 3 \times 7 = 21 \)
- So, (3, 7) is a factor pair.
4. Try 4:
- \( 4 \times \text{(no integer)} = 21 \)
- 4 is not a factor of 21.
5. Continue this process until the square root of 21 (approximately 4.58):
- Since we have already checked up to 4, we can stop here.
The factor pairs of 21 are (1, 21) and (3, 7).
### Answer: (1, 21) and (3, 7)
Expected answer: [(1, 21), (3, 7)]
Correct? Yes
Problem: Could you determine all the factor pairs of 72?
GPT-4o:
To determine all the factor pairs of 72, we need to find all pairs of integers that multiply together to give 72. 1. Start with 1: - \(1 \times 72 = 72\) - Factor pair: (1, 72) 2. Next, try 2: - \(2 \times 36 = 72\) - Factor pair: (2, 36) 3. Next, try 3: - \(3 \times 24 = 72\) - Factor pair: (3, 24) 4. Next, try 4: - \(4 \times 18 = 72\) - Factor pair: (4, 18) 5. Next, try 6: - \(6 \times 12 = 72\) - Factor pair: (6, 12) 6. Next, try 8: - \(8 \times 9 = 72\) - Factor pair: (8, 9) Since 8 is the square root of 72 rounded down, we have found all factor pairs. ### Answer: (1, 72), (2, 36), (3, 24), (4, 18), (6, 12), (8, 9)
Expected answer: [(1, 72), (2, 36), (3, 24), (4, 18), (6, 12), (8, 9)]
Correct? Yes
Problem: Determine all the factor pairs of 38.
GPT-4o:
To determine the factor pairs of 38, we need to find all pairs of integers that multiply together to give 38.
1. Start with 1:
\[
1 \times 38 = 38
\]
So, (1, 38) is a factor pair.
2. Next, check 2:
\[
2 \times 19 = 38
\]
So, (2, 19) is a factor pair.
3. Check 3:
\[
38 \div 3 \approx 12.67 \quad \text{(not an integer, so 3 is not a factor)}
\]
4. Check 4:
\[
38 \div 4 = 9.5 \quad \text{(not an integer, so 4 is not a factor)}
\]
5. Check 5:
\[
38 \div 5 = 7.6 \quad \text{(not an integer, so 5 is not a factor)}
\]
6. Check 6:
\[
38 \div 6 \approx 6.33 \quad \text{(not an integer, so 6 is not a factor)}
\]
7. Check 7:
\[
38 \div 7 \approx 5.43 \quad \text{(not an integer, so 7 is not a factor)}
\]
8. Check 8:
\[
38 \div 8 = 4.75 \quad \text{(not an integer, so 8 is not a factor)}
\]
9. Check 9:
\[
38 \div 9 \approx 4.22 \quad \text{(not an integer, so 9 is not a factor)}
\]
10. Check 10:
\[
38 \div 10 = 3.8 \quad \text{(not an integer, so 10 is not a factor)}
\]
Since we have checked all integers up to the square root of 38 (approximately 6.16), we have found all factor pairs.
### Answer: (1, 38) and (2, 19)
Expected answer: [(1, 38), (2, 19)]
Correct? Yes
Problem: List all the factor pairs of 94.
GPT-4o:
To find the factor pairs of 94, we need to identify all pairs of integers that multiply together to give 94.
1. Start with 1:
\[
1 \times 94 = 94
\]
So, (1, 94) is a factor pair.
2. Check 2:
\[
2 \times 47 = 94
\]
So, (2, 47) is a factor pair.
3. Check 3:
\[
94 \div 3 \approx 31.33 \quad \text{(not an integer, so 3 is not a factor)}
\]
4. Check 4:
\[
94 \div 4 = 23.5 \quad \text{(not an integer, so 4 is not a factor)}
\]
5. Check 5:
\[
94 \div 5 = 18.8 \quad \text{(not an integer, so 5 is not a factor)}
\]
6. Check 6:
\[
94 \div 6 \approx 15.67 \quad \text{(not an integer, so 6 is not a factor)}
\]
7. Check 7:
\[
94 \div 7 \approx 13.43 \quad \text{(not an integer, so 7 is not a factor)}
\]
8. Check 8:
\[
94 \div 8 = 11.75 \quad \text{(not an integer, so 8 is not a factor)}
\]
9. Check 9:
\[
94 \div 9 \approx 10.44 \quad \text{(not an integer, so 9 is not a factor)}
\]
10. Check 10:
\[
94 \div 10 = 9.4 \quad \text{(not an integer, so 10 is not a factor)}
\]
Since we have checked all integers up to the square root of 94 (approximately 9.7), we have found all factor pairs.
### Answer: (1, 94) and (2, 47)
Expected answer: [(1, 94), (2, 47)]
Correct? Yes
Problem: List all factor pairs of 32.
GPT-4o:
To find all factor pairs of 32, we need to identify all pairs of integers that multiply together to give 32. 1. Start with 1: \[ 1 \times 32 = 32 \] So, (1, 32) is a factor pair. 2. Next, try 2: \[ 2 \times 16 = 32 \] So, (2, 16) is a factor pair. 3. Next, try 4: \[ 4 \times 8 = 32 \] So, (4, 8) is a factor pair. 4. Next, try 8: \[ 8 \times 4 = 32 \] This is the same as (4, 8), so we have already listed it. 5. Next, try 16: \[ 16 \times 2 = 32 \] This is the same as (2, 16), so we have already listed it. 6. Finally, try 32: \[ 32 \times 1 = 32 \] This is the same as (1, 32), so we have already listed it. Thus, the factor pairs of 32 are: (1, 32), (2, 16), and (4, 8). ### Answer: (1, 32), (2, 16), (4, 8)
Expected answer: [(1, 32), (2, 16), (4, 8)]
Correct? Yes
Problem: Identify all factor pairs of the number 33.
GPT-4o:
To identify all factor pairs of the number 33, we need to find pairs of integers that multiply together to give 33. 1. Start with 1: - \(1 \times 33 = 33\) - So, (1, 33) is a factor pair. 2. Next, check 2: - \(2 \times 16.5 = 33\) - Since 16.5 is not an integer, (2, 16.5) is not a factor pair. 3. Check 3: - \(3 \times 11 = 33\) - So, (3, 11) is a factor pair. 4. Check 4: - \(4 \times 8.25 = 33\) - Since 8.25 is not an integer, (4, 8.25) is not a factor pair. 5. Check 5: - \(5 \times 6.6 = 33\) - Since 6.6 is not an integer, (5, 6.6) is not a factor pair. 6. Check 6: - \(6 \times 5.5 = 33\) - Since 5.5 is not an integer, (6, 5.5) is not a factor pair. 7. Check 7: - \(7 \times 4.714 = 33\) - Since 4.714 is not an integer, (7, 4.714) is not a factor pair. 8. Check 8 and higher: - Any higher number will result in a product that is less than 33 when multiplied by another integer greater than 1. Thus, the factor pairs of 33 are: - (1, 33) - (3, 11) ### Answer: (1, 33) and (3, 11)
Expected answer: [(1, 33), (3, 11)]
Correct? Yes
Problem: Write down all the factor pairs of 59.
GPT-4o:
To find the factor pairs of 59, we need to identify all pairs of integers that multiply together to give 59. First, we check if 59 is a prime number. A prime number has no divisors other than 1 and itself. We test divisibility by all prime numbers less than the square root of 59 (approximately 7.68). - 59 is not divisible by 2 (it's odd). - 59 is not divisible by 3 (sum of digits, 5 + 9 = 14, is not divisible by 3). - 59 is not divisible by 5 (it does not end in 0 or 5). - 59 is not divisible by 7 (59 ÷ 7 ≈ 8.4286, not an integer). Since 59 is not divisible by any of these primes, it is a prime number. Therefore, the only factor pairs of 59 are (1, 59) and (59, 1). ### Answer: (1, 59) and (59, 1)
Expected answer: [(1, 59)]
Correct? Yes
Problem: List all the factor pairs of the number 54.
GPT-4o:
To find all the factor pairs of 54, we need to identify all pairs of integers that multiply together to give 54. 1. Start with 1: \( 1 \times 54 = 54 \) So, (1, 54) is a factor pair. 2. Next, try 2: \( 2 \times 27 = 54 \) So, (2, 27) is a factor pair. 3. Next, try 3: \( 3 \times 18 = 54 \) So, (3, 18) is a factor pair. 4. Next, try 4: \( 4 \times 13.5 = 54 \) Since 13.5 is not an integer, (4, 13.5) is not a factor pair. 5. Next, try 5: \( 5 \times 10.8 = 54 \) Since 10.8 is not an integer, (5, 10.8) is not a factor pair. 6. Next, try 6: \( 6 \times 9 = 54 \) So, (6, 9) is a factor pair. 7. Next, try 7: \( 7 \times 7.714 \approx 54 \) Since 7.714 is not an integer, (7, 7.714) is not a factor pair. Since we have tried all integers up to the square root of 54 (approximately 7.35), we have found all the factor pairs. ### Answer: (1, 54), (2, 27), (3, 18), (6, 9)
Expected answer: [(1, 54), (2, 27), (3, 18), (6, 9)]
Correct? Yes
Problem: Determine all factor pairs of the number 85.
GPT-4o:
To determine the factor pairs of the number 85, we need to find all pairs of integers that multiply together to give 85.
1. Start with the number 1:
\[
1 \times 85 = 85
\]
So, (1, 85) is a factor pair.
2. Next, check the number 2:
\[
85 \div 2 = 42.5 \quad \text{(not an integer, so 2 is not a factor)}
\]
3. Check the number 3:
\[
85 \div 3 \approx 28.33 \quad \text{(not an integer, so 3 is not a factor)}
\]
4. Check the number 4:
\[
85 \div 4 = 21.25 \quad \text{(not an integer, so 4 is not a factor)}
\]
5. Check the number 5:
\[
5 \times 17 = 85
\]
So, (5, 17) is a factor pair.
6. Check the number 6:
\[
85 \div 6 \approx 14.17 \quad \text{(not an integer, so 6 is not a factor)}
\]
7. Continue checking numbers up to the square root of 85 (approximately 9.22):
- 7: \( 85 \div 7 \approx 12.14 \quad \text{(not an integer, so 7 is not a factor)} \)
- 8: \( 85 \div 8 \approx 10.625 \quad \text{(not an integer, so 8 is not a factor)} \)
- 9: \( 85 \div 9 \approx 9.44 \quad \text{(not an integer, so 9 is not a factor)} \)
Since we have checked all numbers up to the square root of 85, we have found all factor pairs.
### Answer: (1, 85) and (5, 17)
Expected answer: [(1, 85), (5, 17)]
Correct? Yes
Problem: Identify all the factor pairs of the number 56.
GPT-4o:
To identify all the factor pairs of the number 56, we need to find pairs of numbers that, when multiplied together, equal 56.
1. Start with 1:
\[
1 \times 56 = 56
\]
So, (1, 56) is a factor pair.
2. Next, try 2:
\[
2 \times 28 = 56
\]
So, (2, 28) is a factor pair.
3. Next, try 3:
\[
3 \times \text{(not an integer)} \neq 56
\]
So, 3 is not a factor.
4. Next, try 4:
\[
4 \times 14 = 56
\]
So, (4, 14) is a factor pair.
5. Next, try 5:
\[
5 \times \text{(not an integer)} \neq 56
\]
So, 5 is not a factor.
6. Next, try 6:
\[
6 \times \text{(not an integer)} \neq 56
\]
So, 6 is not a factor.
7. Next, try 7:
\[
7 \times 8 = 56
\]
So, (7, 8) is a factor pair.
Since 7 and 8 are close to each other, we have found all the factor pairs.
### Answer: (1, 56), (2, 28), (4, 14), (7, 8)
Expected answer: [(1, 56), (2, 28), (4, 14), (7, 8)]
Correct? Yes
Problem: List all the factor pairs of the number 24.
GPT-4o:
To find all the factor pairs of the number 24, we need to identify all pairs of integers that multiply together to give 24.
1. Start with 1:
- \(1 \times 24 = 24\)
- Factor pair: (1, 24)
2. Next, try 2:
- \(2 \times 12 = 24\)
- Factor pair: (2, 12)
3. Next, try 3:
- \(3 \times 8 = 24\)
- Factor pair: (3, 8)
4. Next, try 4:
- \(4 \times 6 = 24\)
- Factor pair: (4, 6)
5. Next, try 5:
- \(5 \times \text{(not an integer)} = 24\)
- No factor pair here.
6. Next, try 6:
- \(6 \times 4 = 24\)
- This pair is already listed as (4, 6).
Since we have already listed all unique pairs, we can stop here.
### Answer: (1, 24), (2, 12), (3, 8), (4, 6)
Expected answer: [(1, 24), (2, 12), (3, 8), (4, 6)]
Correct? Yes
Problem: What are all the factor pairs of the number 84?
GPT-4o:
To find all the factor pairs of the number 84, we need to identify pairs of numbers that, when multiplied together, equal 84. 1. Start with 1: \[ 1 \times 84 = 84 \] So, (1, 84) is a factor pair. 2. Next, try 2: \[ 2 \times 42 = 84 \] So, (2, 42) is a factor pair. 3. Next, try 3: \[ 3 \times 28 = 84 \] So, (3, 28) is a factor pair. 4. Next, try 4: \[ 4 \times 21 = 84 \] So, (4, 21) is a factor pair. 5. Next, try 6: \[ 6 \times 14 = 84 \] So, (6, 14) is a factor pair. 6. Next, try 7: \[ 7 \times 12 = 84 \] So, (7, 12) is a factor pair. 7. Finally, try 9: \[ 9 \times 9.33 \neq 84 \] So, 9 is not a factor. We have now tested all integers up to the square root of 84 (approximately 9.17). Therefore, the factor pairs of 84 are: ### Answer: (1, 84), (2, 42), (3, 28), (4, 21), (6, 14), (7, 12)
Expected answer: [(1, 84), (2, 42), (3, 28), (4, 21), (6, 14), (7, 12)]
Correct? Yes
Problem: Identify all factor pairs of the number 36.
GPT-4o:
To identify all factor pairs of the number 36, we need to find pairs of numbers that, when multiplied together, equal 36. 1. Start with 1: - \(1 \times 36 = 36\) - Factor pair: (1, 36) 2. Next, try 2: - \(2 \times 18 = 36\) - Factor pair: (2, 18) 3. Next, try 3: - \(3 \times 12 = 36\) - Factor pair: (3, 12) 4. Next, try 4: - \(4 \times 9 = 36\) - Factor pair: (4, 9) 5. Next, try 6: - \(6 \times 6 = 36\) - Factor pair: (6, 6) Since 6 is repeated, we have found all unique factor pairs. ### Answer: (1, 36), (2, 18), (3, 12), (4, 9), (6, 6)
Expected answer: [(1, 36), (2, 18), (3, 12), (4, 9), (6, 6)]
Correct? Yes
Problem: What are all the factor pairs of 85?
GPT-4o:
To find the factor pairs of 85, we need to identify all pairs of integers that multiply together to give 85.
1. Start with 1:
\[
1 \times 85 = 85
\]
So, (1, 85) is a factor pair.
2. Next, check 2:
\[
85 \div 2 = 42.5 \quad \text{(not an integer, so 2 is not a factor)}
\]
3. Check 3:
\[
85 \div 3 \approx 28.33 \quad \text{(not an integer, so 3 is not a factor)}
\]
4. Check 4:
\[
85 \div 4 = 21.25 \quad \text{(not an integer, so 4 is not a factor)}
\]
5. Check 5:
\[
5 \times 17 = 85
\]
So, (5, 17) is a factor pair.
6. Check 6 through 16:
\[
85 \div 6 \approx 14.17 \quad \text{(not an integer, so 6 is not a factor)}
\]
\[
85 \div 7 \approx 12.14 \quad \text{(not an integer, so 7 is not a factor)}
\]
\[
85 \div 8 \approx 10.625 \quad \text{(not an integer, so 8 is not a factor)}
\]
\[
85 \div 9 \approx 9.44 \quad \text{(not an integer, so 9 is not a factor)}
\]
\[
85 \div 10 = 8.5 \quad \text{(not an integer, so 10 is not a factor)}
\]
\[
85 \div 11 \approx 7.73 \quad \text{(not an integer, so 11 is not a factor)}
\]
\[
85 \div 12 \approx 7.08 \quad \text{(not an integer, so 12 is not a factor)}
\]
\[
85 \div 13 \approx 6.54 \quad \text{(not an integer, so 13 is not a factor)}
\]
\[
85 \div 14 \approx 6.07 \quad \text{(not an integer, so 14 is not a factor)}
\]
\[
85 \div 15 \approx 5.67 \quad \text{(not an integer, so 15 is not a factor)}
\]
\[
85 \div 16 \approx 5.31 \quad \text{(not an integer, so 16 is not a factor)}
\]
Since we have checked all integers up to the square root of 85 (approximately 9.22), we have found all factor pairs.
### Answer: (1, 85) and (5, 17)
Expected answer: [(1, 85), (5, 17)]
Correct? Yes
Problem: Identify all factor pairs of the number 34.
GPT-4o:
To identify all factor pairs of the number 34, we need to find pairs of integers that multiply together to give 34.
1. Start with the number 1:
- \(1 \times 34 = 34\)
- So, (1, 34) is a factor pair.
2. Next, check the number 2:
- \(2 \times 17 = 34\)
- So, (2, 17) is a factor pair.
3. Check the number 3:
- \(3 \times \text{(no integer)} = 34\)
- 3 is not a factor of 34.
4. Check the number 4:
- \(4 \times \text{(no integer)} = 34\)
- 4 is not a factor of 34.
5. Continue this process up to the square root of 34 (approximately 5.83):
- Since 5 is not a factor of 34, we stop here.
Thus, the factor pairs of 34 are:
- (1, 34)
- (2, 17)
### Answer: (1, 34) and (2, 17)
Expected answer: [(1, 34), (2, 17)]
Correct? Yes
Problem: Determine all factor pairs of 88.
GPT-4o:
To determine all factor pairs of 88, we need to find all pairs of integers that multiply together to give 88.
1. Start with 1:
\[
1 \times 88 = 88
\]
So, (1, 88) is a factor pair.
2. Next, try 2:
\[
2 \times 44 = 88
\]
So, (2, 44) is a factor pair.
3. Next, try 3:
\[
88 \div 3 \approx 29.33 \quad \text{(not an integer, so 3 is not a factor)}
\]
4. Next, try 4:
\[
4 \times 22 = 88
\]
So, (4, 22) is a factor pair.
5. Next, try 5:
\[
88 \div 5 = 17.6 \quad \text{(not an integer, so 5 is not a factor)}
\]
6. Next, try 6:
\[
88 \div 6 \approx 14.67 \quad \text{(not an integer, so 6 is not a factor)}
\]
7. Next, try 7:
\[
88 \div 7 \approx 12.57 \quad \text{(not an integer, so 7 is not a factor)}
\]
8. Next, try 8:
\[
8 \times 11 = 88
\]
So, (8, 11) is a factor pair.
9. Next, try 9:
\[
88 \div 9 \approx 9.78 \quad \text{(not an integer, so 9 is not a factor)}
\]
10. Finally, try 10:
\[
88 \div 10 = 8.8 \quad \text{(not an integer, so 10 is not a factor)}
\]
Since we have tried all integers up to the square root of 88 (approximately 9.38), we have found all factor pairs.
### Answer: (1, 88), (2, 44), (4, 22), (8, 11)
Expected answer: [(1, 88), (2, 44), (4, 22), (8, 11)]
Correct? Yes
Problem: Write down all factor pairs for the number 83.
GPT-4o:
To find the factor pairs of 83, we need to determine which pairs of integers multiply together to give 83. First, we check if 83 is a prime number. A prime number has no divisors other than 1 and itself. We test divisibility by all prime numbers less than the square root of 83 (approximately 9.1). - 83 is not divisible by 2 (it's odd). - 83 is not divisible by 3 (sum of digits 8 + 3 = 11, which is not divisible by 3). - 83 is not divisible by 5 (it does not end in 0 or 5). - 83 is not divisible by 7 (83 ÷ 7 ≈ 11.857, not an integer). Since 83 is not divisible by any of these primes, it is a prime number. Therefore, the only factor pairs are (1, 83) and (83, 1). ### Answer: (1, 83)
Expected answer: [(1, 83)]
Correct? Yes
Problem: What are all the factor pairs of 60?
GPT-4o:
To find all the factor pairs of 60, we need to identify all pairs of integers that multiply together to give 60.
1. Start with 1:
- \(1 \times 60 = 60\)
- Factor pair: (1, 60)
2. Next, try 2:
- \(2 \times 30 = 60\)
- Factor pair: (2, 30)
3. Next, try 3:
- \(3 \times 20 = 60\)
- Factor pair: (3, 20)
4. Next, try 4:
- \(4 \times 15 = 60\)
- Factor pair: (4, 15)
5. Next, try 5:
- \(5 \times 12 = 60\)
- Factor pair: (5, 12)
6. Next, try 6:
- \(6 \times 10 = 60\)
- Factor pair: (6, 10)
7. Next, try 7:
- \(7 \times \text{(not an integer)}\)
- No factor pair
8. Next, try 8:
- \(8 \times \text{(not an integer)}\)
- No factor pair
9. Next, try 9:
- \(9 \times \text{(not an integer)}\)
- No factor pair
10. Finally, try 10:
- \(10 \times 6 = 60\)
- Factor pair: (10, 6) (already listed as (6, 10))
Thus, the factor pairs of 60 are:
(1, 60), (2, 30), (3, 20), (4, 15), (5, 12), and (6, 10).
### Answer: (1, 60), (2, 30), (3, 20), (4, 15), (5, 12), (6, 10)
Expected answer: [(1, 60), (2, 30), (3, 20), (4, 15), (5, 12), (6, 10)]
Correct? Yes
Problem: Write down all factor pairs of 58.
GPT-4o:
To find the factor pairs of 58, we need to identify all pairs of integers that multiply together to give 58.
1. Start with 1:
\( 1 \times 58 = 58 \)
So, (1, 58) is a factor pair.
2. Check the next integer, 2:
\( 2 \times 29 = 58 \)
So, (2, 29) is a factor pair.
3. Check the next integer, 3:
\( 3 \times \text{(not an integer)} \neq 58 \)
So, 3 is not a factor.
4. Continue checking integers up to the square root of 58 (approximately 7.6):
- 4: \( 4 \times \text{(not an integer)} \neq 58 \)
- 5: \( 5 \times \text{(not an integer)} \neq 58 \)
- 6: \( 6 \times \text{(not an integer)} \neq 58 \)
- 7: \( 7 \times \text{(not an integer)} \neq 58 \)
Since no other integers up to the square root of 58 are factors, we have found all factor pairs.
### Answer: (1, 58) and (2, 29)
Expected answer: [(1, 58), (2, 29)]
Correct? Yes
Problem: Please list all of the factor pairs of the number 40.
GPT-4o:
To find all the factor pairs of the number 40, we need to identify all pairs of integers that multiply together to give 40. 1. Start with 1: - \(1 \times 40 = 40\) - Factor pair: (1, 40) 2. Next, try 2: - \(2 \times 20 = 40\) - Factor pair: (2, 20) 3. Next, try 4: - \(4 \times 10 = 40\) - Factor pair: (4, 10) 4. Next, try 5: - \(5 \times 8 = 40\) - Factor pair: (5, 8) 5. Next, try 8: - \(8 \times 5 = 40\) - This pair is already listed as (5, 8), so we stop here. Thus, the factor pairs of 40 are: - (1, 40) - (2, 20) - (4, 10) - (5, 8) ### Answer: (1, 40), (2, 20), (4, 10), (5, 8)
Expected answer: [(1, 40), (2, 20), (4, 10), (5, 8)]
Correct? Yes
Problem: Identify all the factor pairs of 91.
GPT-4o:
To identify all the factor pairs of 91, we need to find all pairs of integers that multiply to give 91.
1. Start with 1:
\( 1 \times 91 = 91 \)
So, (1, 91) is a factor pair.
2. Check 2:
\( 91 \div 2 = 45.5 \) (not an integer, so 2 is not a factor)
3. Check 3:
\( 91 \div 3 \approx 30.33 \) (not an integer, so 3 is not a factor)
4. Check 4:
\( 91 \div 4 = 22.75 \) (not an integer, so 4 is not a factor)
5. Check 5:
\( 91 \div 5 = 18.2 \) (not an integer, so 5 is not a factor)
6. Check 6:
\( 91 \div 6 \approx 15.17 \) (not an integer, so 6 is not a factor)
7. Check 7:
\( 91 \div 7 = 13 \) (an integer, so 7 is a factor)
So, (7, 13) is a factor pair.
8. Check 8:
\( 91 \div 8 \approx 11.375 \) (not an integer, so 8 is not a factor)
9. Check 9:
\( 91 \div 9 \approx 10.11 \) (not an integer, so 9 is not a factor)
10. Check 10 and higher:
Since 10 and higher numbers up to the square root of 91 (approximately 9.5) have already been checked, we can stop here.
Thus, the factor pairs of 91 are:
(1, 91) and (7, 13).
### Answer: (1, 91) and (7, 13)
Expected answer: [(1, 91), (7, 13)]
Correct? Yes
Problem: List all the factor pairs of 76.
GPT-4o:
To find the factor pairs of 76, we need to identify all pairs of integers that multiply together to give 76.
1. Start with 1:
\[
1 \times 76 = 76
\]
So, (1, 76) is a factor pair.
2. Next, try 2:
\[
2 \times 38 = 76
\]
So, (2, 38) is a factor pair.
3. Next, try 3:
\[
76 \div 3 \approx 25.33 \quad \text{(not an integer, so 3 is not a factor)}
\]
4. Next, try 4:
\[
4 \times 19 = 76
\]
So, (4, 19) is a factor pair.
5. Next, try 5:
\[
76 \div 5 = 15.2 \quad \text{(not an integer, so 5 is not a factor)}
\]
6. Next, try 6:
\[
76 \div 6 \approx 12.67 \quad \text{(not an integer, so 6 is not a factor)}
\]
7. Next, try 7:
\[
76 \div 7 \approx 10.86 \quad \text{(not an integer, so 7 is not a factor)}
\]
8. Next, try 8:
\[
76 \div 8 = 9.5 \quad \text{(not an integer, so 8 is not a factor)}
\]
9. Next, try 9:
\[
76 \div 9 \approx 8.44 \quad \text{(not an integer, so 9 is not a factor)}
\]
10. Next, try 10:
\[
76 \div 10 = 7.6 \quad \text{(not an integer, so 10 is not a factor)}
\]
Since we have checked all integers up to the square root of 76 (approximately 8.7), we have found all factor pairs.
### Answer: (1, 76), (2, 38), (4, 19)
Expected answer: [(1, 76), (2, 38), (4, 19)]
Correct? Yes
Problem: What are all the factor pairs of 13?
GPT-4o:
To find the factor pairs of 13, we need to identify all pairs of integers that multiply together to give 13. First, we note that 13 is a prime number, meaning it has no divisors other than 1 and itself. The factor pairs of 13 are: 1. (1, 13) 2. (13, 1) These are the only pairs of integers that multiply to 13. ### Answer: (1, 13) and (13, 1)
Expected answer: [(1, 13)]
Correct? Yes
Problem: Find all factor pairs of 64.
GPT-4o:
To find all factor pairs of 64, we need to identify all pairs of integers that multiply together to give 64. 1. Start with 1: - \(1 \times 64 = 64\) - Factor pair: (1, 64) 2. Next, try 2: - \(2 \times 32 = 64\) - Factor pair: (2, 32) 3. Next, try 4: - \(4 \times 16 = 64\) - Factor pair: (4, 16) 4. Next, try 8: - \(8 \times 8 = 64\) - Factor pair: (8, 8) Since 8 is the square root of 64, we have now found all unique factor pairs. ### Answer: (1, 64), (2, 32), (4, 16), (8, 8)
Expected answer: [(1, 64), (2, 32), (4, 16), (8, 8)]
Correct? Yes
Problem: What are all factor pairs of 8?
GPT-4o:
To find all factor pairs of 8, we need to determine which pairs of integers multiply together to give the product of 8.
1. Start with 1:
- \(1 \times 8 = 8\)
- So, (1, 8) is a factor pair.
2. Next, try 2:
- \(2 \times 4 = 8\)
- So, (2, 4) is a factor pair.
3. Check 3:
- \(3 \times \text{(no integer)} = 8\)
- 3 is not a factor of 8.
4. We have already reached the square root of 8 (approximately 2.83), so we can stop here.
Thus, the factor pairs of 8 are:
- (1, 8)
- (2, 4)
### Answer: (1, 8) and (2, 4)
Expected answer: [(1, 8), (2, 4)]
Correct? Yes
Problem: List all the factor pairs of 80.
GPT-4o:
To find all the factor pairs of 80, we need to identify all pairs of integers that multiply together to give 80. 1. Start with 1: - \(1 \times 80 = 80\) - Factor pair: (1, 80) 2. Next, try 2: - \(2 \times 40 = 80\) - Factor pair: (2, 40) 3. Next, try 4: - \(4 \times 20 = 80\) - Factor pair: (4, 20) 4. Next, try 5: - \(5 \times 16 = 80\) - Factor pair: (5, 16) 5. Next, try 8: - \(8 \times 10 = 80\) - Factor pair: (8, 10) 6. Check for any other factors: - 3, 6, 7, 9, 11, 12, 13, 14, 15 do not divide 80 evenly. Thus, the factor pairs of 80 are: (1, 80), (2, 40), (4, 20), (5, 16), and (8, 10). ### Answer: (1, 80), (2, 40), (4, 20), (5, 16), (8, 10)
Expected answer: [(1, 80), (2, 40), (4, 20), (5, 16), (8, 10)]
Correct? Yes
Problem: Calculate all the factor pairs for the number 96.
GPT-4o:
To find all the factor pairs of 96, we need to identify all pairs of integers that multiply together to give 96. We start with 1 and go up to the square root of 96, which is approximately 9.8. We will check each integer up to 10 to see if it divides 96 without leaving a remainder. 1. \( 1 \times 96 = 96 \) 2. \( 2 \times 48 = 96 \) 3. \( 3 \times 32 = 96 \) 4. \( 4 \times 24 = 96 \) 5. \( 6 \times 16 = 96 \) 6. \( 8 \times 12 = 96 \) We have checked all integers up to 10, and these are the factor pairs of 96: - (1, 96) - (2, 48) - (3, 32) - (4, 24) - (6, 16) - (8, 12) ### Answer: (1, 96), (2, 48), (3, 32), (4, 24), (6, 16), (8, 12)
Expected answer: [(1, 96), (2, 48), (3, 32), (4, 24), (6, 16), (8, 12)]
Correct? Yes
Problem: Calculate all factor pairs of 19.
GPT-4o:
To find the factor pairs of 19, we need to identify all pairs of integers that multiply together to give 19. First, we note that 19 is a prime number, meaning it has no divisors other than 1 and itself. The factor pairs of 19 are: 1. (1, 19) 2. (19, 1) These are the only pairs of integers that multiply to 19. ### Answer: (1, 19) and (19, 1)
Expected answer: [(1, 19)]
Correct? Yes
Problem: What are all factor pairs of 37?
GPT-4o:
To find the factor pairs of 37, we need to identify all pairs of integers that multiply together to give 37. First, we note that 37 is a prime number, meaning it has no divisors other than 1 and itself. Thus, the only factor pairs of 37 are: 1. (1, 37) 2. (37, 1) These pairs represent the same multiplication, so we only list them once. ### Answer: (1, 37)
Expected answer: [(1, 37)]
Correct? Yes
Problem: Write down all factor pairs of the number 6.
GPT-4o:
To find the factor pairs of the number 6, we need to identify all pairs of integers that multiply together to give 6. 1. Start with 1: - \(1 \times 6 = 6\) - So, (1, 6) is a factor pair. 2. Next, consider 2: - \(2 \times 3 = 6\) - So, (2, 3) is a factor pair. 3. Check 3: - \(3 \times 2 = 6\) - This is the same as the pair (2, 3), so it is not a new pair. 4. Check 6: - \(6 \times 1 = 6\) - This is the same as the pair (1, 6), so it is not a new pair. Thus, the factor pairs of 6 are: - (1, 6) - (2, 3) ### Answer: (1, 6) and (2, 3)
Expected answer: [(1, 6), (2, 3)]
Correct? Yes
Problem: List down all the factor pairs of 58.
GPT-4o:
To find the factor pairs of 58, we need to identify all pairs of integers that multiply together to give 58.
1. Start with 1:
- \(1 \times 58 = 58\)
- So, (1, 58) is a factor pair.
2. Check the next integer, 2:
- \(2 \times 29 = 58\)
- So, (2, 29) is a factor pair.
3. Check the next integer, 3:
- \(3 \times \text{(not an integer)} \neq 58\)
- So, 3 is not a factor.
4. Continue checking integers up to the square root of 58 (approximately 7.6):
- 4: \(4 \times \text{(not an integer)} \neq 58\)
- 5: \(5 \times \text{(not an integer)} \neq 58\)
- 6: \(6 \times \text{(not an integer)} \neq 58\)
- 7: \(7 \times \text{(not an integer)} \neq 58\)
Since no other integers up to the square root of 58 are factors, we have found all the factor pairs.
### Answer: (1, 58) and (2, 29)
Expected answer: [(1, 58), (2, 29)]
Correct? Yes
Problem: List all the factor pairs of the number 81.
GPT-4o:
To find the factor pairs of 81, we need to identify all pairs of integers that multiply together to give 81. 1. Start with 1: - \(1 \times 81 = 81\) - Factor pair: (1, 81) 2. Next, try 2: - 81 is not divisible by 2 (since 81 is odd). 3. Try 3: - \(3 \times 27 = 81\) - Factor pair: (3, 27) 4. Try 4: - 81 is not divisible by 4. 5. Try 5: - 81 is not divisible by 5. 6. Try 6: - 81 is not divisible by 6. 7. Try 7: - 81 is not divisible by 7. 8. Try 8: - 81 is not divisible by 8. 9. Try 9: - \(9 \times 9 = 81\) - Factor pair: (9, 9) Since 9 is the square root of 81, we have found all the factor pairs. The factor pairs of 81 are: (1, 81), (3, 27), and (9, 9). ### Answer: (1, 81), (3, 27), (9, 9)
Expected answer: [(1, 81), (3, 27), (9, 9)]
Correct? Yes
Problem: Please list all factor pairs of the number 85.
GPT-4o:
To find all factor pairs of the number 85, we need to identify all pairs of integers that multiply together to give 85. 1. Start with 1: - \(1 \times 85 = 85\) - So, (1, 85) is a factor pair. 2. Check the next integer, 2: - 85 is not divisible by 2 (since 85 is odd). 3. Check 3: - The sum of the digits of 85 is 8 + 5 = 13, which is not divisible by 3. 4. Check 4: - 85 is not divisible by 4 (since 85 is not even). 5. Check 5: - The last digit of 85 is 5, so 85 is divisible by 5. - \(5 \times 17 = 85\) - So, (5, 17) is a factor pair. 6. Check 6 through 16: - None of these numbers divide 85 evenly. Since we have checked all integers up to the square root of 85 (approximately 9.22), we have found all factor pairs. ### Answer: (1, 85) and (5, 17)
Expected answer: [(1, 85), (5, 17)]
Correct? Yes
Problem: Write down all factor pairs for the number 93.
GPT-4o:
To find all factor pairs of the number 93, we need to identify all pairs of integers that multiply together to give 93. 1. Start with 1: - \(1 \times 93 = 93\) - Factor pair: (1, 93) 2. Check 2: - 93 is not divisible by 2 (since 93 is odd). 3. Check 3: - \(93 \div 3 = 31\) - \(3 \times 31 = 93\) - Factor pair: (3, 31) 4. Check 4: - 93 is not divisible by 4. 5. Check 5: - 93 is not divisible by 5 (since it does not end in 0 or 5). 6. Check 6: - 93 is not divisible by 6. 7. Check 7: - 93 is not divisible by 7. 8. Continue checking up to the square root of 93 (approximately 9.6): - 93 is not divisible by 8 or 9. Since we have checked all possible factors up to the square root of 93, we have found all factor pairs. ### Answer: (1, 93) and (3, 31)
Expected answer: [(1, 93), (3, 31)]
Correct? Yes
Problem: Identify all factor pairs of 24.
GPT-4o:
To identify all factor pairs of 24, we need to find all pairs of integers that multiply together to give 24. 1. Start with 1: - \(1 \times 24 = 24\) - Factor pair: (1, 24) 2. Next, try 2: - \(2 \times 12 = 24\) - Factor pair: (2, 12) 3. Next, try 3: - \(3 \times 8 = 24\) - Factor pair: (3, 8) 4. Next, try 4: - \(4 \times 6 = 24\) - Factor pair: (4, 6) 5. Next, try 5: - 5 is not a factor of 24 since 24 is not divisible by 5. 6. Next, try 6: - \(6 \times 4 = 24\) - This pair is already listed as (4, 6), so we stop here. Thus, the factor pairs of 24 are: (1, 24), (2, 12), (3, 8), and (4, 6). ### Answer: (1, 24), (2, 12), (3, 8), (4, 6)
Expected answer: [(1, 24), (2, 12), (3, 8), (4, 6)]
Correct? Yes
Problem: List all the factor pairs of 97.
GPT-4o:
To find the factor pairs of 97, we need to determine which pairs of integers multiply together to give 97. First, we check if 97 is a prime number. A prime number has no divisors other than 1 and itself. We test divisibility by all prime numbers less than the square root of 97 (approximately 9.8). The prime numbers to test are 2, 3, 5, and 7. - 97 is not even, so it is not divisible by 2. - The sum of the digits of 97 is 9 + 7 = 16, which is not divisible by 3. - 97 does not end in 0 or 5, so it is not divisible by 5. - Dividing 97 by 7 gives approximately 13.857, which is not an integer. Since 97 is not divisible by any of these primes, it is a prime number. Therefore, the only factor pairs of 97 are 1 and 97. ### Answer: (1, 97)
Expected answer: [(1, 97)]
Correct? Yes
Problem: Identify all factor pairs of the number 57.
GPT-4o:
To identify all factor pairs of the number 57, we need to find pairs of integers that multiply together to give 57. 1. Start with 1: - \(1 \times 57 = 57\) - So, (1, 57) is a factor pair. 2. Check 2: - 57 is not divisible by 2 (since 57 is odd). 3. Check 3: - \(57 \div 3 = 19\) - So, \(3 \times 19 = 57\) - (3, 19) is a factor pair. 4. Check 4: - 57 is not divisible by 4. 5. Check 5: - 57 is not divisible by 5. 6. Check 6: - 57 is not divisible by 6. 7. Check 7: - 57 is not divisible by 7. Since 7 is greater than the square root of 57 (approximately 7.55), we can stop here. The factor pairs of 57 are: - (1, 57) - (3, 19) ### Answer: (1, 57) and (3, 19)
Expected answer: [(1, 57), (3, 19)]
Correct? Yes
Problem: List down all the factor pairs of 19.
GPT-4o:
To find the factor pairs of 19, we need to identify all pairs of integers that multiply together to give 19. 1. Start with 1: - \(1 \times 19 = 19\) Since 19 is a prime number, it has no other factors besides 1 and itself. Thus, the factor pairs of 19 are: - (1, 19) ### Answer: (1, 19)
Expected answer: [(1, 19)]
Correct? Yes
Problem: What are all factor pairs of 38?
GPT-4o:
To find all factor pairs of 38, we need to determine which pairs of integers multiply together to give the product of 38.
1. Start with 1:
- \(1 \times 38 = 38\)
- So, (1, 38) is a factor pair.
2. Next, check 2:
- \(2 \times 19 = 38\)
- So, (2, 19) is a factor pair.
3. Check 3:
- \(3 \times \text{(not an integer)} = 38\)
- So, 3 is not a factor.
4. Check 4:
- \(4 \times \text{(not an integer)} = 38\)
- So, 4 is not a factor.
5. Continue this process up to the square root of 38 (approximately 6.16):
- 5 is not a factor because \(5 \times \text{(not an integer)} = 38\)
- 6 is not a factor because \(6 \times \text{(not an integer)} = 38\)
Since we have checked all integers up to the square root of 38, we have found all factor pairs.
### Answer: (1, 38) and (2, 19)
Expected answer: [(1, 38), (2, 19)]
Correct? Yes
Problem: Find all the factor pairs of the number 6.
GPT-4o:
To find all the factor pairs of the number 6, we need to identify all pairs of integers that multiply together to give 6. 1. Start with 1: - \(1 \times 6 = 6\) - So, (1, 6) is a factor pair. 2. Next, try 2: - \(2 \times 3 = 6\) - So, (2, 3) is a factor pair. 3. Check 3: - \(3 \times 2 = 6\) - This is the same as the pair (2, 3), so it is not a new pair. 4. Check 6: - \(6 \times 1 = 6\) - This is the same as the pair (1, 6), so it is not a new pair. Thus, the factor pairs of 6 are (1, 6) and (2, 3). ### Answer: (1, 6) and (2, 3)
Expected answer: [(1, 6), (2, 3)]
Correct? Yes
Problem: Write down all factor pairs of 17.
GPT-4o:
To find the factor pairs of 17, we need to identify all pairs of integers that multiply together to give 17. 1. Start with 1: - \(1 \times 17 = 17\) Since 17 is a prime number, it has no other factors besides 1 and itself. Thus, the factor pairs of 17 are: - (1, 17) ### Answer: (1, 17)
Expected answer: [(1, 17)]
Correct? Yes
Problem: List all factor pairs of 45.
GPT-4o:
To find all factor pairs of 45, we need to identify all pairs of integers that multiply together to give 45.
1. Start with 1:
\( 1 \times 45 = 45 \)
So, (1, 45) is a factor pair.
2. Next, try 2:
\( 2 \times \text{(no integer)} = 45 \)
2 is not a factor of 45.
3. Try 3:
\( 3 \times 15 = 45 \)
So, (3, 15) is a factor pair.
4. Try 4:
\( 4 \times \text{(no integer)} = 45 \)
4 is not a factor of 45.
5. Try 5:
\( 5 \times 9 = 45 \)
So, (5, 9) is a factor pair.
6. Try 6:
\( 6 \times \text{(no integer)} = 45 \)
6 is not a factor of 45.
7. Try 7:
\( 7 \times \text{(no integer)} = 45 \)
7 is not a factor of 45.
8. Try 8:
\( 8 \times \text{(no integer)} = 45 \)
8 is not a factor of 45.
9. Try 9:
\( 9 \times 5 = 45 \)
We already have this pair as (5, 9).
10. Continue until the square root of 45 (approximately 6.7):
We have already covered all possible pairs.
Thus, the factor pairs of 45 are:
(1, 45), (3, 15), and (5, 9).
### Answer: (1, 45), (3, 15), (5, 9)
Expected answer: [(1, 45), (3, 15), (5, 9)]
Correct? Yes
Problem: Write down all factor pairs of 36.
GPT-4o:
To find all factor pairs of 36, we need to identify all pairs of integers that multiply together to give 36. 1. Start with 1: - \(1 \times 36 = 36\) - Factor pair: (1, 36) 2. Next, try 2: - \(2 \times 18 = 36\) - Factor pair: (2, 18) 3. Next, try 3: - \(3 \times 12 = 36\) - Factor pair: (3, 12) 4. Next, try 4: - \(4 \times 9 = 36\) - Factor pair: (4, 9) 5. Next, try 6: - \(6 \times 6 = 36\) - Factor pair: (6, 6) Since 6 is repeated, we have found all unique factor pairs. ### Answer: (1, 36), (2, 18), (3, 12), (4, 9), (6, 6)
Expected answer: [(1, 36), (2, 18), (3, 12), (4, 9), (6, 6)]
Correct? Yes
Problem: What are all factor pairs of 89?
GPT-4o:
To determine the factor pairs of 89, we first need to identify its factors. A factor pair consists of two numbers that, when multiplied together, equal 89. 1. Start by checking if 89 is a prime number. A prime number has no divisors other than 1 and itself. 2. Check divisibility by smaller prime numbers (2, 3, 5, 7): - 89 is odd, so it is not divisible by 2. - The sum of the digits of 89 is 8 + 9 = 17, which is not divisible by 3. - 89 does not end in 0 or 5, so it is not divisible by 5. - For 7, we perform the division: 89 ÷ 7 ≈ 12.71, which is not an integer. Since 89 is not divisible by any of these smaller prime numbers, it is a prime number. Therefore, the only factor pairs of 89 are (1, 89). ### Answer: (1, 89)
Expected answer: [(1, 89)]
Correct? Yes
Problem: List all the factor pairs of 89.
GPT-4o:
To find the factor pairs of 89, we need to determine which pairs of integers multiply together to give 89. First, we note that 89 is a prime number. A prime number has exactly two distinct positive divisors: 1 and itself. Thus, the only factor pairs of 89 are: 1 and 89. ### Answer: (1, 89)
Expected answer: [(1, 89)]
Correct? Yes
Problem: List all the factor pairs of the number 44.
GPT-4o:
To find the factor pairs of the number 44, we need to identify all pairs of integers that multiply together to give 44.
1. Start with 1:
\[
1 \times 44 = 44
\]
So, (1, 44) is a factor pair.
2. Next, try 2:
\[
2 \times 22 = 44
\]
So, (2, 22) is a factor pair.
3. Next, try 3:
\[
3 \times \text{(not an integer)} = 44
\]
3 is not a factor of 44.
4. Next, try 4:
\[
4 \times 11 = 44
\]
So, (4, 11) is a factor pair.
5. Next, try 5:
\[
5 \times \text{(not an integer)} = 44
\]
5 is not a factor of 44.
6. Next, try 6:
\[
6 \times \text{(not an integer)} = 44
\]
6 is not a factor of 44.
7. Continue this process until the square root of 44 (approximately 6.63). Since we have already checked up to 6, we have found all factor pairs.
The factor pairs of 44 are:
\[
(1, 44), (2, 22), (4, 11)
\]
### Answer: (1, 44), (2, 22), (4, 11)
Expected answer: [(1, 44), (2, 22), (4, 11)]
Correct? Yes
Problem: Identify all factor pairs of the number 36.
GPT-4o:
To identify all factor pairs of the number 36, we need to find pairs of numbers that, when multiplied together, equal 36. 1. Start with 1: - \(1 \times 36 = 36\) - Factor pair: (1, 36) 2. Next, try 2: - \(2 \times 18 = 36\) - Factor pair: (2, 18) 3. Next, try 3: - \(3 \times 12 = 36\) - Factor pair: (3, 12) 4. Next, try 4: - \(4 \times 9 = 36\) - Factor pair: (4, 9) 5. Next, try 6: - \(6 \times 6 = 36\) - Factor pair: (6, 6) Since 6 is the square root of 36, we have now found all unique factor pairs. ### Answer: (1, 36), (2, 18), (3, 12), (4, 9), (6, 6)
Expected answer: [(1, 36), (2, 18), (3, 12), (4, 9), (6, 6)]
Correct? Yes
Problem: List all possible factor pairs of the number 35.
GPT-4o:
To find all possible factor pairs of the number 35, we need to identify pairs of integers that, when multiplied together, equal 35. 1. Start with the number 1: - \(1 \times 35 = 35\) - So, (1, 35) is a factor pair. 2. Next, check the number 2: - 35 is not divisible by 2 (since 35 is an odd number). 3. Check the number 3: - 35 is not divisible by 3 (since the sum of the digits, 3 + 5 = 8, is not divisible by 3). 4. Check the number 4: - 35 is not divisible by 4 (since 35 is not an even number). 5. Check the number 5: - \(5 \times 7 = 35\) - So, (5, 7) is a factor pair. 6. Check the number 6: - 35 is not divisible by 6 (since 35 is not an even number and the sum of the digits is not divisible by 3). 7. Check the number 7: - \(7 \times 5 = 35\) - This pair is already listed as (5, 7). Since we have checked all numbers up to the square root of 35 (approximately 5.92), we have found all factor pairs. ### Answer: (1, 35) and (5, 7)
Expected answer: [(1, 35), (5, 7)]
Correct? Yes
Problem: Write down all factor pairs of 22.
GPT-4o:
To find the factor pairs of 22, we need to identify all pairs of integers that multiply together to give 22.
1. Start with 1:
\( 1 \times 22 = 22 \)
So, (1, 22) is a factor pair.
2. Next, try 2:
\( 2 \times 11 = 22 \)
So, (2, 11) is a factor pair.
3. Check 3:
\( 3 \times \text{(no integer)} = 22 \)
3 is not a factor of 22.
4. Check 4:
\( 4 \times \text{(no integer)} = 22 \)
4 is not a factor of 22.
5. Continue this process until the square root of 22 (approximately 4.69):
- 5 is not a factor of 22.
- 6 is not a factor of 22.
- Continue until 4.69.
Since we have checked all possible pairs, the factor pairs of 22 are:
(1, 22) and (2, 11).
### Answer: (1, 22) and (2, 11)
Expected answer: [(1, 22), (2, 11)]
Correct? Yes
Problem: Find all the factor pairs of 40.
GPT-4o:
To find all the factor pairs of 40, we need to identify all pairs of integers that multiply together to give 40.
1. Start with 1:
- \(1 \times 40 = 40\)
- Factor pair: (1, 40)
2. Next, try 2:
- \(2 \times 20 = 40\)
- Factor pair: (2, 20)
3. Next, try 3:
- \(3 \times \text{(not an integer)} = 40\)
- No factor pair for 3
4. Next, try 4:
- \(4 \times 10 = 40\)
- Factor pair: (4, 10)
5. Next, try 5:
- \(5 \times 8 = 40\)
- Factor pair: (5, 8)
6. Next, try 6:
- \(6 \times \text{(not an integer)} = 40\)
- No factor pair for 6
7. Next, try 7:
- \(7 \times \text{(not an integer)} = 40\)
- No factor pair for 7
8. Finally, try 8:
- \(8 \times 5 = 40\)
- This pair is already listed as (5, 8)
We have now checked all integers up to the square root of 40 (approximately 6.32), and we have found all factor pairs.
### Answer: (1, 40), (2, 20), (4, 10), (5, 8)
Expected answer: [(1, 40), (2, 20), (4, 10), (5, 8)]
Correct? Yes
Problem: List all the factor pairs of the number 64.
GPT-4o:
To find all the factor pairs of the number 64, we need to identify all pairs of integers that multiply together to give 64. 1. Start with 1: \[ 1 \times 64 = 64 \] So, (1, 64) is a factor pair. 2. Next, try 2: \[ 2 \times 32 = 64 \] So, (2, 32) is a factor pair. 3. Next, try 4: \[ 4 \times 16 = 64 \] So, (4, 16) is a factor pair. 4. Next, try 8: \[ 8 \times 8 = 64 \] So, (8, 8) is a factor pair. Since 8 is repeated, we do not need to list it twice. Thus, the factor pairs of 64 are: (1, 64), (2, 32), (4, 16), and (8, 8). ### Answer: (1, 64), (2, 32), (4, 16), (8, 8)
Expected answer: [(1, 64), (2, 32), (4, 16), (8, 8)]
Correct? Yes
Problem: Determine all factor pairs of 30.
GPT-4o:
To determine all factor pairs of 30, we need to find all pairs of integers that multiply together to give 30. 1. Start with 1: - \(1 \times 30 = 30\) - Factor pair: (1, 30) 2. Next, try 2: - \(2 \times 15 = 30\) - Factor pair: (2, 15) 3. Next, try 3: - \(3 \times 10 = 30\) - Factor pair: (3, 10) 4. Next, try 4: - \(4 \times 7.5 = 30\) - Since 7.5 is not an integer, (4, 7.5) is not a valid factor pair. 5. Next, try 5: - \(5 \times 6 = 30\) - Factor pair: (5, 6) 6. Next, try 6: - \(6 \times 5 = 30\) - This pair is already listed as (5, 6), so we stop here. Thus, the factor pairs of 30 are: - (1, 30) - (2, 15) - (3, 10) - (5, 6) ### Answer: (1, 30), (2, 15), (3, 10), (5, 6)
Expected answer: [(1, 30), (2, 15), (3, 10), (5, 6)]
Correct? Yes
Problem: List all factor pairs of 59.
GPT-4o:
To find the factor pairs of 59, we need to determine which pairs of integers multiply together to give 59. First, we check if 59 is a prime number. A prime number has no divisors other than 1 and itself. We test divisibility by all prime numbers less than the square root of 59 (approximately 7.68). - 59 is not divisible by 2 (it's odd). - 59 is not divisible by 3 (sum of digits, 5 + 9 = 14, is not divisible by 3). - 59 is not divisible by 5 (it does not end in 0 or 5). - 59 is not divisible by 7 (59 ÷ 7 ≈ 8.4286, not an integer). Since 59 is not divisible by any of these primes, it is a prime number. Therefore, the only factor pairs of 59 are (1, 59). ### Answer: (1, 59)
Expected answer: [(1, 59)]
Correct? Yes
Problem: List all factor pairs of 18.
GPT-4o:
To find all factor pairs of 18, we need to identify all pairs of integers that multiply together to give 18. 1. Start with 1: - \(1 \times 18 = 18\) - Factor pair: (1, 18) 2. Next, try 2: - \(2 \times 9 = 18\) - Factor pair: (2, 9) 3. Next, try 3: - \(3 \times 6 = 18\) - Factor pair: (3, 6) 4. Next, try 4: - \(4 \times 4.5 = 18\) - Since 4.5 is not an integer, (4, 4.5) is not a valid factor pair. 5. Next, try 5: - \(5 \times 3.6 = 18\) - Since 3.6 is not an integer, (5, 3.6) is not a valid factor pair. 6. Finally, try 6: - \(6 \times 3 = 18\) - This is the same as the pair (3, 6), which we already have. Thus, the factor pairs of 18 are: - (1, 18) - (2, 9) - (3, 6) ### Answer: (1, 18), (2, 9), (3, 6)
Expected answer: [(1, 18), (2, 9), (3, 6)]
Correct? Yes
Problem: Please list all the factor pairs of 61.
GPT-4o:
To find the factor pairs of 61, we need to determine which pairs of integers multiply together to give 61. First, we check if 61 is a prime number. A prime number has no divisors other than 1 and itself. We test divisibility by all prime numbers less than the square root of 61 (approximately 7.81). - 61 is not divisible by 2 (it's odd). - 61 is not divisible by 3 (sum of digits, 6 + 1 = 7, is not divisible by 3). - 61 is not divisible by 5 (it does not end in 0 or 5). - 61 is not divisible by 7 (61 ÷ 7 ≈ 8.71, not an integer). Since 61 is not divisible by any of these primes, it is a prime number. Therefore, the only factor pairs of 61 are (1, 61) and (61, 1). ### Answer: (1, 61)
Expected answer: [(1, 61)]
Correct? Yes
Problem: What are all of the factor pairs of 86?
GPT-4o:
To find all the factor pairs of 86, we need to determine which pairs of integers multiply together to give 86. 1. Start with 1: - \(1 \times 86 = 86\) - So, (1, 86) is a factor pair. 2. Check 2: - \(2 \times 43 = 86\) - So, (2, 43) is a factor pair. 3. Check 3: - \(86 \div 3 \approx 28.67\) (not an integer, so 3 is not a factor) 4. Check 4: - \(86 \div 4 = 21.5\) (not an integer, so 4 is not a factor) 5. Check 5: - \(86 \div 5 = 17.2\) (not an integer, so 5 is not a factor) 6. Check 6: - \(86 \div 6 \approx 14.33\) (not an integer, so 6 is not a factor) 7. Check 7: - \(86 \div 7 \approx 12.29\) (not an integer, so 7 is not a factor) 8. Continue this process up to the square root of 86 (approximately 9.3): - 8 and 9 are not factors since \(86 \div 8 \approx 10.75\) and \(86 \div 9 \approx 9.56\) are not integers. Since we have checked all possible factors up to the square root of 86, we have found all the factor pairs. ### Answer: (1, 86) and (2, 43)
Expected answer: [(1, 86), (2, 43)]
Correct? Yes
Problem: List out all factor pairs of 57.
GPT-4o:
To find the factor pairs of 57, we need to identify all pairs of integers that multiply together to give 57.
1. Start with 1:
- \(1 \times 57 = 57\)
- So, (1, 57) is a factor pair.
2. Check the next integer, 2:
- \(2 \times \text{(no integer)} = 57\)
- 57 is not divisible by 2, so 2 is not a factor.
3. Check the next integer, 3:
- \(3 \times 19 = 57\)
- So, (3, 19) is a factor pair.
4. Check the next integer, 4:
- \(4 \times \text{(no integer)} = 57\)
- 57 is not divisible by 4, so 4 is not a factor.
5. Check the next integer, 5:
- \(5 \times \text{(no integer)} = 57\)
- 57 is not divisible by 5, so 5 is not a factor.
6. Check the next integer, 6:
- \(6 \times \text{(no integer)} = 57\)
- 57 is not divisible by 6, so 6 is not a factor.
7. Check the next integer, 7:
- \(7 \times \text{(no integer)} = 57\)
- 57 is not divisible by 7, so 7 is not a factor.
8. Continue this process until the square root of 57 (approximately 7.55). Since we have already checked up to 7, we have found all factor pairs.
The factor pairs of 57 are:
- (1, 57)
- (3, 19)
### Answer: (1, 57) and (3, 19)
Expected answer: [(1, 57), (3, 19)]
Correct? Yes
Problem: What are all factor pairs of 38?
GPT-4o:
To find all factor pairs of 38, we need to determine which pairs of integers multiply together to give the product of 38.
1. Start with 1:
\( 1 \times 38 = 38 \)
So, (1, 38) is a factor pair.
2. Next, check 2:
\( 2 \times 19 = 38 \)
So, (2, 19) is a factor pair.
3. Check 3:
\( 3 \times \text{(not an integer)} \neq 38 \)
So, 3 is not a factor.
4. Check 4:
\( 4 \times \text{(not an integer)} \neq 38 \)
So, 4 is not a factor.
5. Continue this process up to the square root of 38 (approximately 6.16). Since 38 is not divisible by any other integers between 1 and 6, we have found all factor pairs.
The factor pairs of 38 are:
(1, 38) and (2, 19).
### Answer: (1, 38) and (2, 19)
Expected answer: [(1, 38), (2, 19)]
Correct? Yes
Problem: Please list all factor pairs of 27.
GPT-4o:
To find all factor pairs of 27, we need to identify all pairs of integers that multiply together to give 27.
1. Start with 1:
\[
1 \times 27 = 27
\]
So, (1, 27) is a factor pair.
2. Next, try 2:
\[
2 \times \text{(no integer)} = 27
\]
Since 27 is not divisible by 2, there is no factor pair involving 2.
3. Try 3:
\[
3 \times 9 = 27
\]
So, (3, 9) is a factor pair.
4. Try 4:
\[
4 \times \text{(no integer)} = 27
\]
Since 27 is not divisible by 4, there is no factor pair involving 4.
5. Try 5:
\[
5 \times \text{(no integer)} = 27
\]
Since 27 is not divisible by 5, there is no factor pair involving 5.
6. Try 6:
\[
6 \times \text{(no integer)} = 27
\]
Since 27 is not divisible by 6, there is no factor pair involving 6.
7. Finally, try 7 and higher:
\[
7 \times \text{(no integer)} = 27
\]
Since 27 is not divisible by 7, there is no factor pair involving 7.
We have already reached the square root of 27 (approximately 5.2), so we have found all factor pairs.
The factor pairs of 27 are:
\[
(1, 27) \quad \text{and} \quad (3, 9)
\]
### Answer: (1, 27) and (3, 9)
Expected answer: [(1, 27), (3, 9)]
Correct? Yes
Problem: List all the factor pairs of the number 78.
GPT-4o:
To find the factor pairs of 78, we need to identify all pairs of integers that multiply together to give 78. 1. Start with 1: - \(1 \times 78 = 78\) - Factor pair: (1, 78) 2. Next, try 2: - \(2 \times 39 = 78\) - Factor pair: (2, 39) 3. Try 3: - \(3 \times 26 = 78\) - Factor pair: (3, 26) 4. Try 4: - \(4 \times 19.5 \neq 78\) - 4 is not a factor. 5. Try 5: - \(5 \times 15.6 \neq 78\) - 5 is not a factor. 6. Try 6: - \(6 \times 13 = 78\) - Factor pair: (6, 13) 7. Try 7: - \(7 \times 11.14 \neq 78\) - 7 is not a factor. 8. Try 8: - \(8 \times 9.75 \neq 78\) - 8 is not a factor. Since we have tried all integers up to the square root of 78 (approximately 8.83), we have found all factor pairs. ### Answer: (1, 78), (2, 39), (3, 26), (6, 13)
Expected answer: [(1, 78), (2, 39), (3, 26), (6, 13)]
Correct? Yes
Problem: List out all the factor pairs of 23.
GPT-4o:
To find the factor pairs of 23, we need to identify all pairs of integers that multiply together to give 23. 1. Start with 1: - \(1 \times 23 = 23\) Since 23 is a prime number, it has no other factors besides 1 and itself. Thus, the factor pairs of 23 are: - (1, 23) ### Answer: (1, 23)
Expected answer: [(1, 23)]
Correct? Yes
Problem: List down all the factor pairs of 2.
GPT-4o:
To find the factor pairs of 2, we need to identify all pairs of integers that, when multiplied together, result in 2. 1. Start with 1: - \(1 \times 2 = 2\) - This gives us the factor pair (1, 2). 2. Check for any other pairs: - Since 2 is a prime number, it has no other factor pairs besides (1, 2). Thus, the only factor pair of 2 is (1, 2). ### Answer: (1, 2)
Expected answer: [(1, 2)]
Correct? Yes
Problem: Write down all the factor pairs of 49.
GPT-4o:
To find the factor pairs of 49, we need to identify all pairs of integers that multiply together to give 49.
1. Start with 1:
\[
1 \times 49 = 49
\]
So, (1, 49) is a factor pair.
2. Next, check 2:
\[
2 \times \text{(no integer)} = 49
\]
Since 49 is not divisible by 2, 2 is not a factor.
3. Check 3:
\[
3 \times \text{(no integer)} = 49
\]
Since 49 is not divisible by 3, 3 is not a factor.
4. Check 4:
\[
4 \times \text{(no integer)} = 49
\]
Since 49 is not divisible by 4, 4 is not a factor.
5. Check 5:
\[
5 \times \text{(no integer)} = 49
\]
Since 49 is not divisible by 5, 5 is not a factor.
6. Check 6:
\[
6 \times \text{(no integer)} = 49
\]
Since 49 is not divisible by 6, 6 is not a factor.
7. Check 7:
\[
7 \times 7 = 49
\]
So, (7, 7) is a factor pair.
Since we have checked all integers up to the square root of 49 (which is 7), we have found all factor pairs.
### Answer: (1, 49) and (7, 7)
Expected answer: [(1, 49), (7, 7)]
Correct? Yes
Problem: Identify all the distinct pairs of factors that can multiply to give the number 93.
GPT-4o:
To identify all the distinct pairs of factors that multiply to give the number 93, we need to find pairs of integers \((a, b)\) such that \(a \times b = 93\). First, we perform the prime factorization of 93: - 93 is an odd number, so it is not divisible by 2. - The sum of the digits of 93 is \(9 + 3 = 12\), which is divisible by 3. Therefore, 93 is divisible by 3. - Dividing 93 by 3, we get \(93 \div 3 = 31\). Thus, the prime factorization of 93 is \(3 \times 31\). Now, we list the pairs of factors: 1. \(1 \times 93 = 93\) 2. \(3 \times 31 = 93\) These are the distinct pairs of factors of 93. ### Answer: (1, 93) and (3, 31)
Expected answer: [(1, 93), (3, 31)]
Correct? Yes