Claude-3 Opus 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.
Claude-3 Opus:
To find all factor pairs of 64, we need to find all pairs of numbers that, when multiplied together, equal 64. Let's start with the smallest possible factors and work our way up. Factors of 64: 1 x 64 = 64 2 x 32 = 64 4 x 16 = 64 8 x 8 = 64 Since we have already found the pair 8 x 8, all other factor pairs will be duplicates of the ones we have already found. Therefore, the factor pairs of 64 are: (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.
Claude-3 Opus:
To find the factor pairs of 18, we need to find all the pairs of numbers that, when multiplied together, give 18. Factors of 18: 1 x 18 = 18 2 x 9 = 18 3 x 6 = 18 Therefore, the factor pairs of 18 are: (1, 18) (2, 9) (3, 6) Note that the order of the numbers in each pair doesn't matter, as multiplication is commutative. For example, (2, 9) and (9, 2) are considered the same factor pair. ### Answer: The factor pairs of 18 are (1, 18), (2, 9), and (3, 6).
Expected answer: [(1, 18), (2, 9), (3, 6)]
Correct? Yes
Problem: Find all factor pairs of 28.
Claude-3 Opus:
To find all factor pairs of 28, we need to find all pairs of integers that, when multiplied together, result in 28. Let's start with the smallest possible factors and work our way up. Factors of 28: 1 x 28 = 28 2 x 14 = 28 4 x 7 = 28 Since we have reached the square root of 28 (which is between 5 and 6), we have found all the factor pairs. The factors of 28 are always paired as follows: - 1 and 28 - 2 and 14 - 4 and 7 Therefore, the factor pairs of 28 are (1, 28), (2, 14), and (4, 7). ### 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.
Claude-3 Opus:
To find all factor pairs of 48, we need to find all pairs of integers that, when multiplied together, result in 48. The factors of 48 are: 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48. The factor pairs of 48 are: 1 × 48 = 48 2 × 24 = 48 3 × 16 = 48 4 × 12 = 48 6 × 8 = 48 Therefore, the factor pairs of 48 are: (1, 48) (2, 24) (3, 16) (4, 12) (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.
Claude-3 Opus:
To find the factor pairs of 7, we need to find all pairs of integers that, when multiplied together, result in 7. The only factor pair of 7 is: 1 and 7 This is because: 1 × 7 = 7 There are no other integer pairs that multiply to give 7. ### Answer: The factor pairs of 7 are (1, 7).
Expected answer: [(1, 7)]
Correct? Yes
Problem: Write down every pair of factors of 98.
Claude-3 Opus:
To find the factors of 98, let's divide 98 by each integer from 1 to 98 and see which divisions result in whole numbers. Factors of 98: 1 x 98 = 98 2 x 49 = 98 7 x 14 = 98 Therefore, the pairs of factors of 98 are: (1, 98) (2, 49) (7, 14) Note that the order of the numbers in each pair doesn't matter, as multiplication is commutative. ### 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.
Claude-3 Opus:
To find the factor pairs of 26, we need to find all pairs of integers that, when multiplied together, result in 26. Factors of 26: 1 × 26 = 26 2 × 13 = 26 Therefore, the factor pairs of 26 are: (1, 26) and (2, 13) ### Answer: (1, 26) and (2, 13)
Expected answer: [(1, 26), (2, 13)]
Correct? Yes
Problem: Write down all factor pairs of 59.
Claude-3 Opus:
To find all factor pairs of 59, we need to find all pairs of numbers that, when multiplied together, equal 59. 59 is a prime number, which means it has only two factors: 1 and itself. Therefore, the only factor pair of 59 is: 1 × 59 = 59 ### Answer: The only factor pair of 59 is (1, 59).
Expected answer: [(1, 59)]
Correct? Yes
Problem: Find out all factor pairs of 20.
Claude-3 Opus:
To find all factor pairs of 20, let's list out all the factors of 20 and then pair them up. Factors of 20: 1 (always a factor of any number) 2 (20 ÷ 2 = 10, no remainder) 4 (20 ÷ 4 = 5, no remainder) 5 (20 ÷ 5 = 4, no remainder) 10 (20 ÷ 10 = 2, no remainder) 20 (always a factor of itself) Now, let's pair these factors: 1 and 20 (1 × 20 = 20) 2 and 10 (2 × 10 = 20) 4 and 5 (4 × 5 = 20) Therefore, the factor pairs of 20 are: (1, 20) (2, 10) (4, 5) ### Answer: The factor pairs of 20 are (1, 20), (2, 10), and (4, 5).
Expected answer: [(1, 20), (2, 10), (4, 5)]
Correct? Yes
Problem: Write down all factor pairs of 20.
Claude-3 Opus:
To find all factor pairs of 20, let's find all the numbers that divide evenly into 20. Factors of 20: 1 x 20 = 20 2 x 10 = 20 4 x 5 = 20 Therefore, the factor pairs of 20 are: (1, 20) (2, 10) (4, 5) Note that the order of the numbers in each pair doesn't matter, so (1, 20) and (20, 1) are considered the same factor pair. ### 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.
Claude-3 Opus:
To find the factor pairs of 13, we need to find all pairs of numbers that, when multiplied together, give 13. Factors of 13: 1 × 13 = 13 13 × 1 = 13 Since 13 is a prime number, it has only two factors: 1 and 13 itself. Therefore, the factor pairs of 13 are: (1, 13) and (13, 1) ### Answer: The factor pairs of 13 are (1, 13) and (13, 1).
Expected answer: [(1, 13)]
Correct? Yes
Problem: Find all factor pairs of 81.
Claude-3 Opus:
To find all factor pairs of 81, let's break it down: 1) The factors of 81 are the numbers that divide evenly into 81. 2) The factor pairs will be the pairs of numbers that when multiplied together equal 81. 3) To find the factors, we can start with 1 and 81, then work our way inward. 1 x 81 = 81 3 x 27 = 81 (because 81 ÷ 3 = 27) 9 x 9 = 81 (because 81 ÷ 9 = 9) 4) Therefore, the factor pairs of 81 are: (1, 81) (3, 27) (9, 9) ### Answer: The factor pairs of 81 are (1, 81), (3, 27), and (9, 9).
Expected answer: [(1, 81), (3, 27), (9, 9)]
Correct? Yes
Problem: List all factor pairs of 76.
Claude-3 Opus:
To find the factor pairs of 76, we need to find all pairs of integers that, when multiplied together, result in 76. Factors of 76: 1 x 76 = 76 2 x 38 = 76 4 x 19 = 76 Therefore, the factor pairs of 76 are: (1, 76) (2, 38) (4, 19) Note that the order of the numbers in each pair doesn't matter, as multiplication is commutative. For example, (1, 76) and (76, 1) are considered the same factor pair. ### 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.
Claude-3 Opus:
To find all factor pairs of 50, we need to find all pairs of integers that, when multiplied together, equal 50. Let's start with the smallest possible factors and work our way up. 1 x 50 = 50 2 x 25 = 50 5 x 10 = 50 Since 50 is not a perfect square, there are no other factor pairs. The factors of 50 are 1, 2, 5, 10, 25, and 50. Therefore, the factor pairs of 50 are: (1, 50) (2, 25) (5, 10) ### Answer: The factor pairs of 50 are (1, 50), (2, 25), and (5, 10).
Expected answer: [(1, 50), (2, 25), (5, 10)]
Correct? Yes
Problem: List all the factor pairs of the number 76.
Claude-3 Opus:
To find the factor pairs of 76, we need to find all pairs of integers that, when multiplied together, result in 76. Factors of 76: 1 × 76 = 76 2 × 38 = 76 4 × 19 = 76 Therefore, the factor pairs of 76 are: (1, 76) (2, 38) (4, 19) Note that the order of the numbers in each pair doesn't matter, as multiplication is commutative. ### Answer: The factor pairs of 76 are (1, 76), (2, 38), and (4, 19).
Expected answer: [(1, 76), (2, 38), (4, 19)]
Correct? Yes
Problem: List all factor pairs of 56.
Claude-3 Opus:
To find the factor pairs of 56, we need to find all pairs of numbers that, when multiplied together, equal 56. Factors of 56: 1 x 56 = 56 2 x 28 = 56 4 x 14 = 56 7 x 8 = 56 Therefore, the factor pairs of 56 are: (1, 56) (2, 28) (4, 14) (7, 8) ### 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.
Claude-3 Opus:
To find the factor pairs of 44, we need to find all pairs of integers that, when multiplied together, equal 44. The factors of 44 are: 1 x 44 = 44 2 x 22 = 44 4 x 11 = 44 Therefore, the factor pairs of 44 are: (1, 44) (2, 22) (4, 11) Note that the order of the numbers in each pair doesn't matter, as multiplication is commutative. For example, (1, 44) and (44, 1) are considered the same factor pair. ### Answer: The factor pairs of 44 are (1, 44), (2, 22), and (4, 11).
Expected answer: [(1, 44), (2, 22), (4, 11)]
Correct? Yes
Problem: List all the factor pairs of 5.
Claude-3 Opus:
To find the factor pairs of 5, we need to find all the pairs of numbers that, when multiplied together, equal 5. The factors of 5 are: 1 and 5 Therefore, the factor pairs of 5 are: (1, 5) and (5, 1) Note that (1, 5) and (5, 1) are considered the same factor pair, just in a different order. ### Answer: (1, 5) and (5, 1)
Expected answer: [(1, 5)]
Correct? Yes
Problem: Identify all factor pairs of the number 95.
Claude-3 Opus:
To find the factor pairs of 95, we need to find all pairs of integers that, when multiplied together, result in 95. Let's go through the possible factors systematically. Factors of 95: 1 and 95: 1 × 95 = 95 5 and 19: 5 × 19 = 95 There are no other integer pairs that multiply to give 95 because 95 is not divisible by 2, 3, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, or 18. Therefore, the factor pairs of 95 are: (1, 95) and (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.
Claude-3 Opus:
To find the factor pairs of 36, we need to find all the pairs of numbers that, when multiplied together, give 36. Factors of 36: 1 x 36 = 36 2 x 18 = 36 3 x 12 = 36 4 x 9 = 36 6 x 6 = 36 Therefore, the factor pairs of 36 are: (1, 36) (2, 18) (3, 12) (4, 9) (6, 6) ### Answer: The factor pairs of 36 are (1, 36), (2, 18), (3, 12), (4, 9), and (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.
Claude-3 Opus:
To find all factor pairs of 59, we need to find all pairs of numbers that, when multiplied together, equal 59. Let's start by listing out the factors of 59: 1 and 59 are factors of 59 because 1 × 59 = 59. Since 59 is a prime number, it has no other factors besides 1 and itself. Therefore, the only factor pair of 59 is: (1, 59) ### Answer: (1, 59)
Expected answer: [(1, 59)]
Correct? Yes
Problem: What are all the factor pairs of 3?
Claude-3 Opus:
To find the factor pairs of 3, we need to find all the pairs of integers that, when multiplied together, give the product of 3. The factors of 3 are: 1 and 3 -1 and -3 This is because: 1 × 3 = 3 -1 × -3 = 3 Therefore, the factor pairs of 3 are: (1, 3) and (-1, -3) ### Answer: The factor pairs of 3 are (1, 3) and (-1, -3).
Expected answer: [(1, 3)]
Correct? Yes
Problem: Could you list all factor pairs of 51?
Claude-3 Opus:
Let's think about this step-by-step: 1) The factors of a number are the numbers that divide into it evenly (with no remainder). 2) 51 is an odd number, so it can't be divided evenly by 2. 3) Let's check the other numbers from 1 to 51: 1 × 51 = 51 3 × 17 = 51 4) There are no other numbers that divide into 51 evenly. 5) Therefore, the factor pairs of 51 are (1, 51) and (3, 17). ### Answer: The factor pairs of 51 are (1, 51) and (3, 17).
Expected answer: [(1, 51), (3, 17)]
Correct? Yes
Problem: List all factor pairs of 50.
Claude-3 Opus:
To find all factor pairs of 50, we need to find all pairs of integers that, when multiplied together, equal 50. Factors of 50: 1 × 50 = 50 2 × 25 = 50 5 × 10 = 50 Therefore, 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.
Claude-3 Opus:
To find the factor pairs of 47, we need to find all pairs of numbers that, when multiplied together, equal 47. 47 is a prime number, which means it has exactly two factors: 1 and itself. Therefore, the only factor pair of 47 is: 1 × 47 = 47 ### Answer: The factor pairs of 47 are (1, 47).
Expected answer: [(1, 47)]
Correct? Yes
Problem: Write down all factor pairs of 51.
Claude-3 Opus:
Let's think through this step-by-step: 1) The factors of a number are the numbers that divide into it evenly (i.e., with no remainder). 2) The number 51 is the product of two prime numbers: 3 and 17. 3) The factor pairs of a number are the pairs of numbers that, when multiplied together, give the original number. 4) Since 51 is the product of two prime numbers, it has only two factors: 1 and itself. 5) Therefore, the only factor pair of 51 is 1 and 51. ### Answer: The factor pairs of 51 are: (1, 51).
Expected answer: [(1, 51), (3, 17)]
Correct? No
Problem: List all factor pairs of 60.
Claude-3 Opus:
To find all factor pairs of 60, we need to find all pairs of integers that, when multiplied together, result in 60. Let's start with the smallest possible factors and work our way up. Factors of 60: 1 x 60 = 60 2 x 30 = 60 3 x 20 = 60 4 x 15 = 60 5 x 12 = 60 6 x 10 = 60 Therefore, the factor pairs of 60 are: (1, 60) (2, 30) (3, 20) (4, 15) (5, 12) (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.
Claude-3 Opus:
To find the factor pairs of 43, we need to find all pairs of numbers that, when multiplied together, result in 43. Let's start with 1 and 43: 1 × 43 = 43 Now, let's check if there are any other numbers between 1 and 43 that divide evenly into 43. 2 does not divide evenly into 43 (43 ÷ 2 = 21.5) 3 does not divide evenly into 43 (43 ÷ 3 = 14.333...) 4 does not divide evenly into 43 (43 ÷ 4 = 10.75) ... 42 does not divide evenly into 43 (43 ÷ 42 = 1.024...) After checking all the numbers from 2 to 42, we find that none of them divide evenly into 43. This means that 43 has only one factor pair. The factor pairs of 43 are: 1 and 43 Since 43 has only two factors (1 and itself), it is a prime number. ### Answer: The factor pairs of 43 are: 1 and 43.
Expected answer: [(1, 43)]
Correct? Yes
Problem: List all factor pairs of 70.
Claude-3 Opus:
To find the factor pairs of 70, we need to find all pairs of integers that, when multiplied together, result in 70. Factors of 70: 1 x 70 = 70 2 x 35 = 70 5 x 14 = 70 7 x 10 = 70 Therefore, the factor pairs of 70 are: (1, 70) (2, 35) (5, 14) (7, 10) ### 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.
Claude-3 Opus:
The factors of a number are the positive integers that divide into the number without leaving a remainder. For the number 2, the factors are: 1 and 2 This is because: 1 × 2 = 2 2 × 1 = 2 No other positive integer divides 2 without leaving a remainder. ### Answer: The factors of 2 are 1 and 2.
Expected answer: [(1, 2)]
Correct? Yes
Problem: List all the factor pairs of 49.
Claude-3 Opus:
To find the factor pairs of 49, we need to find all the pairs of numbers that, when multiplied together, equal 49. Factors of 49: 1 x 49 = 49 7 x 7 = 49 Therefore, the factor pairs of 49 are: (1, 49) (7, 7) ### Answer: (1, 49) and (7, 7)
Expected answer: [(1, 49), (7, 7)]
Correct? Yes
Problem: Identify all factor pairs of 17.
Claude-3 Opus:
To find the factor pairs of 17, we need to find all pairs of integers that, when multiplied together, result in 17. Let's start with 1 and 17: 1 × 17 = 17 Now, let's consider other possible factors between 1 and 17: 2 is not a factor of 17 because 17 is not divisible by 2 (it's an odd number). 3 is not a factor of 17 because 17 is not divisible by 3. 4 is not a factor of 17 because 17 is not divisible by 4. ... 16 is not a factor of 17 because 17 is not divisible by 16. Therefore, the only factor pair of 17 is: (1, 17) ### Answer: The only factor pair of 17 is (1, 17).
Expected answer: [(1, 17)]
Correct? Yes
Problem: Find all the factor pairs of 35.
Claude-3 Opus:
To find all the factor pairs of 35, we need to find all the pairs of numbers that, when multiplied together, give 35. Factors of 35: 1 x 35 = 35 5 x 7 = 35 Therefore, the factor pairs of 35 are: (1, 35) (5, 7) Note that the order of the numbers in each pair doesn't matter, as multiplication is commutative. For example, (1, 35) and (35, 1) are considered the same factor pair. ### Answer: (1, 35) and (5, 7)
Expected answer: [(1, 35), (5, 7)]
Correct? Yes
Problem: Identify all factor pairs of the number 91.
Claude-3 Opus:
To find the factor pairs of 91, we need to find all pairs of integers that, when multiplied together, result in 91. Let's start by finding the factors of 91. Factors of 91: 1 and 91 (since 1 × 91 = 91) 7 and 13 (since 7 × 13 = 91) Therefore, the factor pairs of 91 are: (1, 91) (7, 13) Note that the order of the numbers in each pair doesn't matter, as multiplication is commutative. For example, (1, 91) and (91, 1) are considered the same factor pair. ### Answer: The factor pairs of 91 are (1, 91) and (7, 13).
Expected answer: [(1, 91), (7, 13)]
Correct? Yes
Problem: List all factor pairs of the number 97.
Claude-3 Opus:
To find the factor pairs of 97, we need to find all pairs of numbers that, when multiplied together, result in 97. Factors of 97: 1 × 97 = 97 97 ÷ 1 = 97 and 97 ÷ 97 = 1, so 1 and 97 are factors of 97. Since 97 is a prime number, it has no other factors besides 1 and itself. Therefore, the only factor pair of 97 is (1, 97). ### Answer: (1, 97)
Expected answer: [(1, 97)]
Correct? Yes
Problem: What are all the factor pairs of 16?
Claude-3 Opus:
To find all the factor pairs of 16, let's find all the numbers that divide evenly into 16. Factors of 16: 1 × 16 = 16 2 × 8 = 16 4 × 4 = 16 Therefore, the factor pairs of 16 are: (1, 16) (2, 8) (4, 4) Note that (4, 4) is a factor pair because 4 is a factor of 16 twice. ### Answer: The factor pairs of 16 are (1, 16), (2, 8), and (4, 4).
Expected answer: [(1, 16), (2, 8), (4, 4)]
Correct? Yes
Problem: Find all the factor pairs of the number 21.
Claude-3 Opus:
To find all the factor pairs of 21, we need to find all pairs of integers that, when multiplied together, result in 21. Factors of 21: 1 x 21 = 21 3 x 7 = 21 Therefore, the factor pairs of 21 are: (1, 21) (3, 7) Note that the order of the numbers in each pair doesn't matter, as multiplication is commutative. For example, (1, 21) and (21, 1) are considered the same factor pair. ### Answer: (1, 21) and (3, 7)
Expected answer: [(1, 21), (3, 7)]
Correct? Yes
Problem: Could you determine all the factor pairs of 72?
Claude-3 Opus:
Sure, let's find all the factor pairs of 72. Factors are numbers that divide evenly into 72. To find the factor pairs, we can start with the smallest possible pair and work our way up. 1 x 72 = 72 2 x 36 = 72 3 x 24 = 72 4 x 18 = 72 6 x 12 = 72 8 x 9 = 72 Therefore, the factor pairs of 72 are: (1, 72) (2, 36) (3, 24) (4, 18) (6, 12) (8, 9) Note that the order of the numbers in each pair doesn't matter, as multiplication is commutative. ### Answer: The factor pairs of 72 are (1, 72), (2, 36), (3, 24), (4, 18), (6, 12), and (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.
Claude-3 Opus:
To find all the factor pairs of 38, we need to find all pairs of integers that, when multiplied together, result in 38. We can start by finding the factors of 38 and then pairing them up. Factors of 38: 1 and 38 2 and 19 Therefore, 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: List all the factor pairs of 94.
Claude-3 Opus:
To find the factor pairs of 94, we need to find all pairs of integers that, when multiplied together, result in 94. Factors of 94: 1 x 94 = 94 2 x 47 = 94 Since 94 is an even number, we can check if it's divisible by other even numbers like 4 and 6. 4 is not a factor of 94 because 94 ÷ 4 = 23.5, which is not an integer. 6 is not a factor of 94 because 94 ÷ 6 = 15.666..., which is not an integer. We can also check if 94 is divisible by prime numbers like 3, 5, 7, 11, etc. 3 is not a factor of 94 because 94 ÷ 3 = 31.333..., which is not an integer. 5 is not a factor of 94 because 94 ÷ 5 = 18.8, which is not an integer. 7 is not a factor of 94 because 94 ÷ 7 = 13.428..., which is not an integer. 11 is not a factor of 94 because 94 ÷ 11 = 8.545..., which is not an integer. Since we have checked all possible factors up to the square root of 94 (which is approximately 9.7), we can conclude that there are no more factor pairs. Therefore, the factor pairs of 94 are: 1 and 94 2 and 47 ### Answer: The factor pairs of 94 are (1, 94) and (2, 47).
Expected answer: [(1, 94), (2, 47)]
Correct? Yes
Problem: List all factor pairs of 32.
Claude-3 Opus:
To find all factor pairs of 32, let's find all the numbers that divide evenly into 32. Factors of 32: 1 x 32 = 32 2 x 16 = 32 4 x 8 = 32 Therefore, the factor pairs of 32 are: (1, 32) (2, 16) (4, 8) Note that the order of the numbers in each pair doesn't matter, so (32, 1) is the same as (1, 32). ### 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.
Claude-3 Opus:
To find the factor pairs of 33, we need to find all pairs of integers that, when multiplied together, result in 33. Factors of 33: 1 x 33 = 33 3 x 11 = 33 Therefore, the factor pairs of 33 are: (1, 33) and (3, 11) Note that the order of the numbers in each pair doesn't matter, as multiplication is commutative. ### Answer: The factor pairs of 33 are (1, 33) and (3, 11).
Expected answer: [(1, 33), (3, 11)]
Correct? Yes
Problem: Write down all the factor pairs of 59.
Claude-3 Opus:
To find the factor pairs of 59, we need to find all the pairs of numbers that, when multiplied together, give 59. 59 is a prime number, which means it has only two factors: 1 and itself. Therefore, the factor pairs of 59 are: 1 × 59 = 59 ### Answer: The factor pairs of 59 are (1, 59).
Expected answer: [(1, 59)]
Correct? Yes
Problem: List all the factor pairs of the number 54.
Claude-3 Opus:
To find the factor pairs of 54, we need to find all pairs of numbers that, when multiplied together, equal 54. Factors of 54: 1 x 54 = 54 2 x 27 = 54 3 x 18 = 54 6 x 9 = 54 Therefore, the factor pairs of 54 are: (1, 54) (2, 27) (3, 18) (6, 9) ### 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.
Claude-3 Opus:
To find all factor pairs of 85, we need to find all pairs of integers that, when multiplied together, result in 85. Let's start by finding the factors of 85 and then pairing them up. Factors of 85: 1 and 85 5 and 17 Therefore, the factor pairs of 85 are: (1, 85) and (5, 17) ### Answer: (1, 85) and (5, 17)
Expected answer: [(1, 85), (5, 17)]
Correct? Yes
Problem: Identify all the factor pairs of the number 56.
Claude-3 Opus:
To find the factor pairs of 56, we need to find all the pairs of numbers that, when multiplied together, give 56. Factors of 56: 1 x 56 = 56 2 x 28 = 56 4 x 14 = 56 7 x 8 = 56 Therefore, the factor pairs of 56 are: (1, 56) (2, 28) (4, 14) (7, 8) ### Answer: The factor pairs of 56 are (1, 56), (2, 28), (4, 14), and (7, 8).
Expected answer: [(1, 56), (2, 28), (4, 14), (7, 8)]
Correct? Yes
Problem: List all the factor pairs of the number 24.
Claude-3 Opus:
To find the factor pairs of 24, we need to find all the pairs of numbers that, when multiplied together, equal 24. The factors of 24 are: 1, 2, 3, 4, 6, 8, 12, and 24. The factor pairs of 24 are: 1 × 24 = 24 2 × 12 = 24 3 × 8 = 24 4 × 6 = 24 Therefore, the factor pairs of 24 are: (1, 24) (2, 12) (3, 8) (4, 6) ### 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?
Claude-3 Opus:
To find all the factor pairs of 84, let's find all the numbers that divide evenly into 84. Factors of 84: 1 x 84 = 84 2 x 42 = 84 3 x 28 = 84 4 x 21 = 84 6 x 14 = 84 7 x 12 = 84 Therefore, the factor pairs of 84 are: (1, 84) (2, 42) (3, 28) (4, 21) (6, 14) (7, 12) ### Answer: The factor pairs of 84 are (1, 84), (2, 42), (3, 28), (4, 21), (6, 14), and (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.
Claude-3 Opus:
To find the factor pairs of 36, we need to find all pairs of integers that, when multiplied together, result in 36. Factors of 36: 1 × 36 = 36 2 × 18 = 36 3 × 12 = 36 4 × 9 = 36 6 × 6 = 36 Therefore, the factor pairs of 36 are: (1, 36) (2, 18) (3, 12) (4, 9) (6, 6) ### Answer: The factor pairs of 36 are (1, 36), (2, 18), (3, 12), (4, 9), and (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?
Claude-3 Opus:
To find all the factor pairs of 85, we need to find all the pairs of numbers that, when multiplied together, give 85. Let's start with the smallest possible factors and work our way up. Factors of 85: 1 x 85 = 85 5 x 17 = 85 Therefore, the factor pairs of 85 are: (1, 85) and (5, 17) ### Answer: (1, 85) and (5, 17)
Expected answer: [(1, 85), (5, 17)]
Correct? Yes
Problem: Identify all factor pairs of the number 34.
Claude-3 Opus:
To find the factor pairs of 34, we need to find all pairs of integers that, when multiplied together, result in 34. Factors of 34: 1 x 34 = 34 2 x 17 = 34 Therefore, the factor pairs of 34 are: (1, 34) and (2, 17) ### Answer: (1, 34) and (2, 17)
Expected answer: [(1, 34), (2, 17)]
Correct? Yes
Problem: Determine all factor pairs of 88.
Claude-3 Opus:
To find all factor pairs of 88, we need to find all pairs of integers that, when multiplied together, result in 88. We can start by finding the factors of 88 and then pairing them up. Factors of 88: 1 and 88 2 and 44 4 and 22 8 and 11 Therefore, the factor pairs of 88 are: (1, 88) (2, 44) (4, 22) (8, 11) Note that the order of the numbers in each pair doesn't matter, as multiplication is commutative. ### Answer: The factor pairs of 88 are (1, 88), (2, 44), (4, 22), and (8, 11).
Expected answer: [(1, 88), (2, 44), (4, 22), (8, 11)]
Correct? Yes
Problem: Write down all factor pairs for the number 83.
Claude-3 Opus:
To find the factor pairs of 83, we need to find all pairs of numbers that, when multiplied together, equal 83. Let's start with the smallest possible factor pair and work our way up: 1 x 83 = 83 Since 83 is a prime number, it has no other factors besides 1 and itself. Therefore, the only factor pair for 83 is: (1, 83) ### Answer: (1, 83)
Expected answer: [(1, 83)]
Correct? Yes
Problem: What are all the factor pairs of 60?
Claude-3 Opus:
To find all the factor pairs of 60, let's find all the numbers that divide evenly into 60. Factors of 60: 1 x 60 = 60 2 x 30 = 60 3 x 20 = 60 4 x 15 = 60 5 x 12 = 60 6 x 10 = 60 Therefore, the factor pairs of 60 are: (1, 60) (2, 30) (3, 20) (4, 15) (5, 12) (6, 10) ### Answer: The factor pairs of 60 are (1, 60), (2, 30), (3, 20), (4, 15), (5, 12), and (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.
Claude-3 Opus:
To find all factor pairs of 58, we need to find all pairs of integers that, when multiplied together, result in 58. Let's start with 1 and work our way up. 1 x 58 = 58 2 x 29 = 58 Since 58 is not divisible by 3, 4, 5, 6, 7, 8, or 9, there are no more factor pairs. Therefore, the factor pairs of 58 are: (1, 58) (2, 29) ### Answer: (1, 58), (2, 29)
Expected answer: [(1, 58), (2, 29)]
Correct? Yes
Problem: Please list all of the factor pairs of the number 40.
Claude-3 Opus:
To find the factor pairs of 40, we need to find all pairs of integers that, when multiplied together, result in 40. Factors of 40: 1 x 40 = 40 2 x 20 = 40 4 x 10 = 40 5 x 8 = 40 Therefore, the factor pairs of 40 are: (1, 40) (2, 20) (4, 10) (5, 8) We can also include the negative factor pairs: (-1, -40) (-2, -20) (-4, -10) (-5, -8) ### Answer: The factor pairs of 40 are (1, 40), (2, 20), (4, 10), (5, 8), and their negative counterparts.
Expected answer: [(1, 40), (2, 20), (4, 10), (5, 8)]
Correct? Yes
Problem: Identify all the factor pairs of 91.
Claude-3 Opus:
To find the factor pairs of 91, we need to find all pairs of integers that, when multiplied together, result in 91. Let's start by finding the factors of 91. Factors of 91: 1 and 91 (always a factor pair for any integer) 7 and 13 (because 7 × 13 = 91) Therefore, the factor pairs of 91 are: (1, 91) (7, 13) Note that the order of the numbers in each pair doesn't matter, as multiplication is commutative. For example, (1, 91) and (91, 1) are considered the same factor pair. ### Answer: The factor pairs of 91 are (1, 91) and (7, 13).
Expected answer: [(1, 91), (7, 13)]
Correct? Yes
Problem: List all the factor pairs of 76.
Claude-3 Opus:
To find the factor pairs of 76, we need to find all pairs of integers that, when multiplied together, result in 76. Factors of 76: 1 x 76 = 76 2 x 38 = 76 4 x 19 = 76 Therefore, the factor pairs of 76 are: (1, 76) (2, 38) (4, 19) Note that the order of the numbers in each pair doesn't matter, as multiplication is commutative. ### Answer: The factor pairs of 76 are (1, 76), (2, 38), and (4, 19).
Expected answer: [(1, 76), (2, 38), (4, 19)]
Correct? Yes
Problem: What are all the factor pairs of 13?
Claude-3 Opus:
To find the factor pairs of 13, we need to find all the pairs of numbers that, when multiplied together, give 13. Let's start with 1 and 13: 1 × 13 = 13 Now, let's check if there are any numbers between 1 and 13 that divide 13 evenly (with no remainder). 2 does not divide 13 evenly (13 ÷ 2 = 6 remainder 1) 3 does not divide 13 evenly (13 ÷ 3 = 4 remainder 1) 4 does not divide 13 evenly (13 ÷ 4 = 3 remainder 1) ... 12 does not divide 13 evenly (13 ÷ 12 = 1 remainder 1) Since no other numbers between 1 and 13 divide 13 evenly, there are no other factor pairs. Therefore, the only factor pair of 13 is: 1 and 13 ### Answer: The only factor pair of 13 is 1 and 13.
Expected answer: [(1, 13)]
Correct? Yes
Problem: Find all factor pairs of 64.
Claude-3 Opus:
To find all factor pairs of 64, we need to find all pairs of numbers that, when multiplied together, equal 64. Let's start with the smallest possible factors and work our way up. Factors of 64: 1 x 64 = 64 2 x 32 = 64 4 x 16 = 64 8 x 8 = 64 Since we have already found the pair 8 x 8, all other factor pairs will be duplicates of the ones we have already found. Therefore, the factor pairs of 64 are: (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: What are all factor pairs of 8?
Claude-3 Opus:
To find all factor pairs of 8, let's find all the numbers that divide evenly into 8. Factors of 8: 1 × 8 = 8 2 × 4 = 8 Therefore, the factor pairs of 8 are: (1, 8) and (2, 4) Note that the order of the numbers in each pair doesn't matter, so (1, 8) and (8, 1) are considered the same factor pair. ### Answer: The factor pairs of 8 are (1, 8) and (2, 4).
Expected answer: [(1, 8), (2, 4)]
Correct? Yes
Problem: List all the factor pairs of 80.
Claude-3 Opus:
To find the factor pairs of 80, we need to find all the pairs of numbers that, when multiplied together, give 80. The factors of 80 are: 1 x 80 = 80 2 x 40 = 80 4 x 20 = 80 5 x 16 = 80 8 x 10 = 80 Therefore, the factor pairs of 80 are: (1, 80) (2, 40) (4, 20) (5, 16) (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.
Claude-3 Opus:
To find all the factor pairs for 96, we need to find all the pairs of numbers that, when multiplied together, equal 96. Factors of 96: 1 x 96 = 96 2 x 48 = 96 3 x 32 = 96 4 x 24 = 96 6 x 16 = 96 8 x 12 = 96 Therefore, the factor pairs of 96 are: (1, 96) (2, 48) (3, 32) (4, 24) (6, 16) (8, 12) ### Answer: The factor pairs of 96 are (1, 96), (2, 48), (3, 32), (4, 24), (6, 16), and (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.
Claude-3 Opus:
To find all factor pairs of 19, we need to find all pairs of integers that, when multiplied together, result in 19. Factors of 19: 1 × 19 = 19 19 × 1 = 19 Since 19 is a prime number, it has only two factors: 1 and itself. Therefore, the only factor pair of 19 is (1, 19). ### Answer: (1, 19)
Expected answer: [(1, 19)]
Correct? Yes
Problem: What are all factor pairs of 37?
Claude-3 Opus:
To find all factor pairs of 37, we need to find all pairs of numbers that, when multiplied together, equal 37. Let's start by listing out the factors of 37: 1 and 37 Since 37 is a prime number, it only has two factors: 1 and itself. Therefore, there is only one factor pair for 37: (1, 37) ### Answer: The only factor pair of 37 is (1, 37).
Expected answer: [(1, 37)]
Correct? Yes
Problem: Write down all factor pairs of the number 6.
Claude-3 Opus:
To find all factor pairs of 6, let's find all the pairs of numbers that, when multiplied together, equal 6. Factors of 6: 1 × 6 = 6 2 × 3 = 6 Therefore, 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: List down all the factor pairs of 58.
Claude-3 Opus:
To find the factor pairs of 58, we need to find all pairs of integers that, when multiplied together, result in 58. Factors of 58: 1 x 58 = 58 2 x 29 = 58 Since 58 is not divisible by any other integer besides 1, 2, 29, and 58, there are no other factor pairs. Therefore, the factor pairs of 58 are: (1, 58) (2, 29) ### Answer: (1, 58) and (2, 29)
Expected answer: [(1, 58), (2, 29)]
Correct? Yes
Problem: List all the factor pairs of the number 81.
Claude-3 Opus:
To find the factor pairs of 81, we need to find all the pairs of numbers that, when multiplied together, equal 81. Factors of 81: 1 x 81 = 81 3 x 27 = 81 9 x 9 = 81 Therefore, the factor pairs of 81 are: (1, 81) (3, 27) (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.
Claude-3 Opus:
To find the factor pairs of 85, we need to find all pairs of numbers that, when multiplied together, result in 85. Factors of 85: 1 × 85 = 85 5 × 17 = 85 Therefore, the factor pairs of 85 are: (1, 85) (5, 17) ### Answer: The factor pairs of 85 are (1, 85) and (5, 17).
Expected answer: [(1, 85), (5, 17)]
Correct? Yes
Problem: Write down all factor pairs for the number 93.
Claude-3 Opus:
To find all factor pairs for the number 93, we need to find all pairs of numbers that, when multiplied together, equal 93. Factors of 93: 1 x 93 = 93 3 x 31 = 93 Therefore, the factor pairs of 93 are: (1, 93) and (3, 31) ### Answer: (1, 93) and (3, 31)
Expected answer: [(1, 93), (3, 31)]
Correct? Yes
Problem: Identify all factor pairs of 24.
Claude-3 Opus:
To find all factor pairs of 24, we need to find all pairs of integers that, when multiplied together, result in 24. Let's start with the smallest possible factors and work our way up. Factors of 24: 1 x 24 = 24 2 x 12 = 24 3 x 8 = 24 4 x 6 = 24 Therefore, the factor pairs of 24 are: (1, 24) (2, 12) (3, 8) (4, 6) Note that the order of the factors in each pair doesn't matter, as multiplication is commutative. ### Answer: The factor pairs of 24 are (1, 24), (2, 12), (3, 8), and (4, 6).
Expected answer: [(1, 24), (2, 12), (3, 8), (4, 6)]
Correct? Yes
Problem: List all the factor pairs of 97.
Claude-3 Opus:
To find the factor pairs of 97, we need to find all the pairs of numbers that, when multiplied together, give 97. Factors of 97: 1 x 97 = 97 Since 97 is a prime number, it has only two factors: 1 and 97. Therefore, there is only one factor pair for 97. Factor pairs of 97: (1, 97) ### Answer: (1, 97)
Expected answer: [(1, 97)]
Correct? Yes
Problem: Identify all factor pairs of the number 57.
Claude-3 Opus:
To find the factor pairs of 57, we need to find all pairs of numbers that, when multiplied together, equal 57. Factors of 57: 1 x 57 = 57 3 x 19 = 57 There are no other whole numbers that can be multiplied together to equal 57 because 57 is not divisible by 2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, or 18. Therefore, the factor pairs of 57 are: (1, 57) and (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.
Claude-3 Opus:
To find the factor pairs of 19, we need to find all the pairs of numbers that, when multiplied together, give 19 as the product. Factors of 19: 1 × 19 = 19 19 × 1 = 19 Since 19 is a prime number, it has only two factors: 1 and 19 itself. Therefore, the factor pairs of 19 are: (1, 19) and (19, 1) ### Answer: (1, 19) and (19, 1)
Expected answer: [(1, 19)]
Correct? Yes
Problem: What are all factor pairs of 38?
Claude-3 Opus:
To find all factor pairs of 38, let's find all the factors of 38 and then pair them up. Factors of 38: 1 is a factor of 38 because 38 ÷ 1 = 38 (no remainder) 2 is a factor of 38 because 38 ÷ 2 = 19 (no remainder) 19 is a factor of 38 because 38 ÷ 19 = 2 (no remainder) 38 is a factor of 38 because 38 ÷ 38 = 1 (no remainder) Now, let's pair up these factors: 1 and 38 2 and 19 Therefore, 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: Find all the factor pairs of the number 6.
Claude-3 Opus:
To find all the factor pairs of the number 6, we need to find all the pairs of numbers that, when multiplied together, give the product of 6. Factors of 6: 1 × 6 = 6 2 × 3 = 6 Therefore, 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.
Claude-3 Opus:
To find the factor pairs of 17, we need to find all pairs of numbers that, when multiplied together, equal 17. The only factor pair of 17 is: 1 × 17 = 17 This is because 17 is a prime number, which means it has exactly two factors: 1 and itself. ### Answer: (1, 17)
Expected answer: [(1, 17)]
Correct? Yes
Problem: List all factor pairs of 45.
Claude-3 Opus:
To find all factor pairs of 45, we need to find all pairs of numbers that, when multiplied together, equal 45. Factors of 45: 1 x 45 = 45 3 x 15 = 45 5 x 9 = 45 Therefore, the factor pairs of 45 are: (1, 45) (3, 15) (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.
Claude-3 Opus:
To find all factor pairs of 36, let's find all the numbers that divide evenly into 36. Factors of 36: 1 x 36 = 36 2 x 18 = 36 3 x 12 = 36 4 x 9 = 36 6 x 6 = 36 Therefore, the factor pairs of 36 are: (1, 36) (2, 18) (3, 12) (4, 9) (6, 6) ### 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?
Claude-3 Opus:
To find all factor pairs of 89, we need to find all pairs of numbers that, when multiplied together, equal 89. 89 is a prime number. The only factors of a prime number are 1 and the number itself. Therefore, the factor pairs of 89 are: 1 × 89 = 89 ### Answer: The factor pairs of 89 are (1, 89).
Expected answer: [(1, 89)]
Correct? Yes
Problem: List all the factor pairs of 89.
Claude-3 Opus:
To find the factor pairs of 89, we need to find all the pairs of numbers that, when multiplied together, give 89. 89 is a prime number, which means it has only two factors: 1 and itself. Therefore, the factor pairs of 89 are: 1 × 89 = 89 ### Answer: The factor pairs of 89 are (1, 89).
Expected answer: [(1, 89)]
Correct? Yes
Problem: List all the factor pairs of the number 44.
Claude-3 Opus:
To find the factor pairs of 44, we need to find all pairs of integers that, when multiplied together, equal 44. The factors of 44 are: 1 x 44 = 44 2 x 22 = 44 4 x 11 = 44 Therefore, the factor pairs of 44 are: (1, 44) (2, 22) (4, 11) Note that the order of the numbers in each pair doesn't matter, as multiplication is commutative. For example, (1, 44) and (44, 1) are considered the same factor pair. ### Answer: The factor pairs of 44 are (1, 44), (2, 22), and (4, 11).
Expected answer: [(1, 44), (2, 22), (4, 11)]
Correct? Yes
Problem: Identify all factor pairs of the number 36.
Claude-3 Opus:
To find the factor pairs of 36, we need to find all pairs of integers that, when multiplied together, result in 36. Factors of 36: 1 × 36 = 36 2 × 18 = 36 3 × 12 = 36 4 × 9 = 36 6 × 6 = 36 Therefore, the factor pairs of 36 are: (1, 36) (2, 18) (3, 12) (4, 9) (6, 6) ### Answer: The factor pairs of 36 are (1, 36), (2, 18), (3, 12), (4, 9), and (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.
Claude-3 Opus:
To find all possible factor pairs of 35, we need to find all pairs of numbers that, when multiplied together, equal 35. Factors of 35: 1 x 35 = 35 5 x 7 = 35 Therefore, the factor pairs of 35 are: (1, 35) (5, 7) Note that the order of the numbers in each pair doesn't matter, as multiplication is commutative. For example, (1, 35) and (35, 1) are considered the same factor pair. ### Answer: (1, 35) and (5, 7)
Expected answer: [(1, 35), (5, 7)]
Correct? Yes
Problem: Write down all factor pairs of 22.
Claude-3 Opus:
To find all factor pairs of 22, we need to find all pairs of integers that, when multiplied together, equal 22. Factors of 22: 1 x 22 = 22 2 x 11 = 22 Therefore, 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.
Claude-3 Opus:
To find all the factor pairs of 40, we need to find all the pairs of integers that, when multiplied together, result in 40. Factors of 40: 1 x 40 = 40 2 x 20 = 40 4 x 10 = 40 5 x 8 = 40 Therefore, the factor pairs of 40 are: (1, 40) (2, 20) (4, 10) (5, 8) We can also include the negative factor pairs: (-1, -40) (-2, -20) (-4, -10) (-5, -8) ### Answer: The factor pairs of 40 are (1, 40), (2, 20), (4, 10), (5, 8), and their negative counterparts.
Expected answer: [(1, 40), (2, 20), (4, 10), (5, 8)]
Correct? Yes
Problem: List all the factor pairs of the number 64.
Claude-3 Opus:
To find the factor pairs of 64, we need to find all the pairs of numbers that, when multiplied together, give 64. Factors of 64: 1 x 64 = 64 2 x 32 = 64 4 x 16 = 64 8 x 8 = 64 Therefore, the factor pairs of 64 are: (1, 64) (2, 32) (4, 16) (8, 8) Note that the factor pairs include the number itself paired with 1, and the square root of the number paired with itself (if it's a perfect square). ### 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.
Claude-3 Opus:
To find all factor pairs of 30, we need to find all pairs of integers that, when multiplied together, result in 30. Let's start with the smallest possible factors and work our way up. Factors of 30: 1 × 30 = 30 2 × 15 = 30 3 × 10 = 30 5 × 6 = 30 Therefore, the factor pairs of 30 are: (1, 30) (2, 15) (3, 10) (5, 6) ### Answer: The factor pairs of 30 are (1, 30), (2, 15), (3, 10), and (5, 6).
Expected answer: [(1, 30), (2, 15), (3, 10), (5, 6)]
Correct? Yes
Problem: List all factor pairs of 59.
Claude-3 Opus:
To find the factor pairs of 59, we need to find all pairs of numbers that, when multiplied together, equal 59. 59 is a prime number, which means it has only two factors: 1 and itself. The factor pairs of 59 are: 1 × 59 = 59 Therefore, the only factor pair of 59 is (1, 59). ### Answer: (1, 59)
Expected answer: [(1, 59)]
Correct? Yes
Problem: List all factor pairs of 18.
Claude-3 Opus:
To find the factor pairs of 18, we need to find all pairs of integers that, when multiplied together, result in 18. The factors of 18 are: 1 x 18 = 18 2 x 9 = 18 3 x 6 = 18 Therefore, 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.
Claude-3 Opus:
To find the factor pairs of 61, we need to find all the pairs of numbers that, when multiplied together, give 61. Let's start with the smallest possible factor pair and work our way up: 1 x 61 = 61 Since 61 is a prime number, it has no other factors besides 1 and itself. Therefore, the only factor pair of 61 is: (1, 61) ### Answer: (1, 61)
Expected answer: [(1, 61)]
Correct? Yes
Problem: What are all of the factor pairs of 86?
Claude-3 Opus:
To find all the factor pairs of 86, let's find all the numbers that divide evenly into 86. Factors of 86: 1 x 86 = 86 2 x 43 = 86 Since 86 is an even number, let's check if it's divisible by other even numbers like 4 and 6. 86 ÷ 4 = 21 remainder 2, so 4 is not a factor of 86. 86 ÷ 6 = 14 remainder 2, so 6 is not a factor of 86. Let's also check if it's divisible by prime numbers less than its square root (√86 ≈ 9.27). 86 ÷ 3 = 28 remainder 2, so 3 is not a factor of 86. 86 ÷ 5 = 17 remainder 1, so 5 is not a factor of 86. 86 ÷ 7 = 12 remainder 2, so 7 is not a factor of 86. Therefore, the factor pairs of 86 are: 1 and 86 2 and 43 ### Answer: (1, 86), (2, 43)
Expected answer: [(1, 86), (2, 43)]
Correct? Yes
Problem: List out all factor pairs of 57.
Claude-3 Opus:
To find the factor pairs of 57, we need to find all pairs of numbers that, when multiplied together, result in 57. Factors of 57: 1 x 57 = 57 3 x 19 = 57 Therefore, the factor pairs of 57 are: (1, 57) (3, 19) Note that 57 is not a perfect square, so it does not have any factor pairs with the same number repeated twice. ### Answer: (1, 57), (3, 19)
Expected answer: [(1, 57), (3, 19)]
Correct? Yes
Problem: What are all factor pairs of 38?
Claude-3 Opus:
To find all factor pairs of 38, let's find all the factors of 38 and then pair them up. Factors of 38: 1 is a factor of 38 because 38 ÷ 1 = 38 (no remainder) 2 is a factor of 38 because 38 ÷ 2 = 19 (no remainder) 19 is a factor of 38 because 38 ÷ 19 = 2 (no remainder) 38 is a factor of 38 because 38 ÷ 38 = 1 (no remainder) Now, let's pair up these factors: 1 and 38 2 and 19 Therefore, 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.
Claude-3 Opus:
To find the factor pairs of 27, we need to find all pairs of integers that, when multiplied together, result in 27. Factors of 27: 1 × 27 = 27 3 × 9 = 27 Therefore, the factor pairs of 27 are: (1, 27) (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.
Claude-3 Opus:
To find the factor pairs of 78, we need to find all pairs of numbers that, when multiplied together, equal 78. Factors of 78: 1 x 78 = 78 2 x 39 = 78 3 x 26 = 78 6 x 13 = 78 Therefore, the factor pairs of 78 are: (1, 78) (2, 39) (3, 26) (6, 13) ### 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.
Claude-3 Opus:
To find the factor pairs of 23, we need to find all the pairs of numbers that, when multiplied together, give 23. Factors of 23: 1 x 23 = 23 23 x 1 = 23 Since 23 is a prime number, it has only two factors: 1 and 23 itself. Therefore, the factor pairs of 23 are: (1, 23) and (23, 1) ### Answer: (1, 23) and (23, 1)
Expected answer: [(1, 23)]
Correct? Yes
Problem: List down all the factor pairs of 2.
Claude-3 Opus:
To find the factor pairs of 2, we need to find all the pairs of integers that, when multiplied together, give the product of 2. Factors of 2: 1 × 2 = 2 2 × 1 = 2 Therefore, the factor pairs of 2 are: (1, 2) and (2, 1) Note that (1, 2) and (2, 1) are considered the same factor pair, as the order of the factors does not matter when finding factor pairs. ### Answer: The factor pairs of 2 are (1, 2) and (2, 1).
Expected answer: [(1, 2)]
Correct? Yes
Problem: Write down all the factor pairs of 49.
Claude-3 Opus:
To find the factor pairs of 49, we need to find all the pairs of numbers that, when multiplied together, give 49. Factors of 49: 1 x 49 = 49 7 x 7 = 49 Therefore, the factor pairs of 49 are: (1, 49) and (7, 7) ### 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.
Claude-3 Opus:
Okay, let's think through this step-by-step: 1) First, let's find all the factors of 93. To do this, we can divide 93 by each number from 1 to 93 and see which divisions result in a whole number. 1 is a factor of 93 because 93 ÷ 1 = 93 3 is a factor of 93 because 93 ÷ 3 = 31 31 is a factor of 93 because 93 ÷ 31 = 3 93 is a factor of 93 because 93 ÷ 93 = 1 So, the factors of 93 are 1, 3, 31, and 93. 2) Now, let's make pairs of these factors. Remember, order doesn't matter in multiplication, so (1, 93) is the same as (93, 1). (1, 93) (3, 31) 3) Therefore, there are 2 distinct pairs of factors that can multiply to give 93: (1, 93) and (3, 31). ### Answer: (1, 93) and (3, 31)
Expected answer: [(1, 93), (3, 31)]
Correct? Yes