Problem: Find all factor pairs of 64.

Gemini-1.5 Flash:

Here's how to find all the factor pairs of 64:

* **Start with 1:** 1 is a factor of every number, and its pair is the number itself (64).  So our first pair is (1, 64).
* **Check divisibility by 2:** 64 is even, so 2 is a factor.  64 / 2 = 32, so our next pair is (2, 32).
* **Check divisibility by 4:** 64 is divisible by 4. 64 / 4 = 16, giving us (4, 16).
* **Check divisibility by 8:** 64 is divisible by 8. 64 / 8 = 8, giving us (8, 8).  Notice that we've reached the same number for both factors.  
* **We're done!**  We don't need to check any larger numbers since we've already found all the pairs.

**Here are all the factor pairs of 64:**

(1, 64)
(2, 32)
(4, 16)
(8, 8) 

Expected answer: [(1, 64), (2, 32), (4, 16), (8, 8)]

Correct? No

Problem: List all the factor pairs of 18.

Gemini-1.5 Flash:

Here are the factor pairs of 18:

* **1 and 18** 
* **2 and 9**
* **3 and 6** 

Expected answer: [(1, 18), (2, 9), (3, 6)]

Correct? No

Problem: Find all factor pairs of 28.

Gemini-1.5 Flash:

Here's how to find all the factor pairs of 28:

**Understanding Factors**

Factors are numbers that divide evenly into another number.  A factor pair is a set of two numbers that multiply to give you the original number.

**Finding the Factor Pairs**

1. **Start with 1:**  1 is always a factor of any number.  The pair with 1 is 28 (1 x 28 = 28).

2. **Check divisibility by 2:** 28 is even, so 2 is a factor. The pair with 2 is 14 (2 x 14 = 28).

3. **Continue checking:**  
   - 3 does not divide evenly into 28.
   - 4 does divide evenly into 28. The pair with 4 is 7 (4 x 7 = 28).

4. **You've reached the middle:** Once you reach a factor that is greater than or equal to the other factor in the pair, you've found all the factor pairs.

**The Factor Pairs of 28:**

* 1 and 28
* 2 and 14
* 4 and 7 

Expected answer: [(1, 28), (2, 14), (4, 7)]

Correct? No

Problem: List all factor pairs of the number 48.

Gemini-1.5 Flash:

Here are the factor pairs of 48:

* **1 and 48**
* **2 and 24**
* **3 and 16**
* **4 and 12**
* **6 and 8** 

Expected answer: [(1, 48), (2, 24), (3, 16), (4, 12), (6, 8)]

Correct? No

Problem: Write down all factor pairs of 7.

Gemini-1.5 Flash:

The only factor pairs of 7 are:

* **1 and 7** 
 
This is because 7 is a prime number, meaning it is only divisible by 1 and itself. 

Expected answer: [(1, 7)]

Correct? No

Problem: Write down every pair of factors of 98.

Gemini-1.5 Flash:

Here are all the pairs of factors of 98:

* **1 and 98**
* **2 and 49**
* **7 and 14** 

Expected answer: [(1, 98), (2, 49), (7, 14)]

Correct? No

Problem: List all the factor pairs of the number 26.

Gemini-1.5 Flash:

Here are the factor pairs of 26:

* **1 and 26** (1 x 26 = 26)
* **2 and 13** (2 x 13 = 26) 

Expected answer: [(1, 26), (2, 13)]

Correct? No

Problem: Write down all factor pairs of 59.

Gemini-1.5 Flash:

The only factor pairs of 59 are:

* **1 and 59** 

This is because 59 is a prime number, meaning it is only divisible by 1 and itself. 

Expected answer: [(1, 59)]

Correct? No

Problem: Find out all factor pairs of 20.

Gemini-1.5 Flash:

Here are all the factor pairs of 20:

* **1 and 20** (1 x 20 = 20)
* **2 and 10** (2 x 10 = 20)
* **4 and 5** (4 x 5 = 20) 

Expected answer: [(1, 20), (2, 10), (4, 5)]

Correct? No

Problem: Write down all factor pairs of 20.

Gemini-1.5 Flash:

Here are all the factor pairs of 20:

* **1 and 20** 
* **2 and 10**
* **4 and 5** 

Expected answer: [(1, 20), (2, 10), (4, 5)]

Correct? No

Problem: Please list down all factor pairs of 13.

Gemini-1.5 Flash:

The only factor pairs of 13 are:

* **1 and 13** 

This is because 13 is a prime number, meaning it is only divisible by 1 and itself. 

Expected answer: [(1, 13)]

Correct? No

Problem: Find all factor pairs of 81.

Gemini-1.5 Flash:

Here's how to find all the factor pairs of 81:

* **Start with 1:**  1 is always a factor of any number. Since 81 x 1 = 81, (1, 81) is a factor pair.

* **Check divisibility by 2:** 81 is not divisible by 2, so there's no factor pair involving 2.

* **Check divisibility by 3:** 81 is divisible by 3 (81 / 3 = 27), so (3, 27) is a factor pair.

* **Continue checking:** 
    * 81 is not divisible by 4
    * 81 is divisible by 9 (81 / 9 = 9), so (9, 9) is a factor pair. 

* **Notice a pattern:**  We've found all the factor pairs:

    * (1, 81)
    * (3, 27)
    * (9, 9) 

Notice that we don't need to check numbers larger than 9 because if a number larger than 9 divides 81, its corresponding factor would be smaller than 9 and we've already found those. 
### Answer: (1, 81), (3, 27), (9, 9) 

Expected answer: [(1, 81), (3, 27), (9, 9)]

Correct? Yes

Problem: List all factor pairs of 76.

Gemini-1.5 Flash:

Here are the factor pairs of 76:

* **1 and 76**
* **2 and 38**
* **4 and 19** 

Let me know if you'd like to explore factors of other numbers! 

Expected answer: [(1, 76), (2, 38), (4, 19)]

Correct? No

Problem: Write down all factor pairs of 50.

Gemini-1.5 Flash:

Here are all the factor pairs of 50:

* **1 and 50** 
* **2 and 25**
* **5 and 10** 

Expected answer: [(1, 50), (2, 25), (5, 10)]

Correct? No

Problem: List all the factor pairs of the number 76.

Gemini-1.5 Flash:

Here are the factor pairs of 76:

* **1 and 76**
* **2 and 38**
* **4 and 19** 
 
Let me know if you'd like to try factoring another number! 

Expected answer: [(1, 76), (2, 38), (4, 19)]

Correct? No

Problem: List all factor pairs of 56.

Gemini-1.5 Flash:

Here are the factor pairs of 56:

* **1 and 56**
* **2 and 28**
* **4 and 14**
* **7 and 8** 

Expected answer: [(1, 56), (2, 28), (4, 14), (7, 8)]

Correct? No

Problem: List all the factor pairs of the number 44.

Gemini-1.5 Flash:

Here are the factor pairs of 44:

* 1 and 44
* 2 and 22
* 4 and 11 

Expected answer: [(1, 44), (2, 22), (4, 11)]

Correct? No

Problem: List all the factor pairs of 5.

Gemini-1.5 Flash:

The only factor pairs of 5 are:

* **1 and 5** 

This is because 5 is a prime number, meaning it is only divisible by 1 and itself. 

Expected answer: [(1, 5)]

Correct? No

Problem: Identify all factor pairs of the number 95.

Gemini-1.5 Flash:

Here's how to find the factor pairs of 95:

* **Start with 1:**  1 is always a factor of any number. Its pair is the number itself: 1 x 95 = 95
* **Check divisibility by other numbers:**  95 is an odd number, so it's not divisible by 2.  It doesn't end in a 0 or 5, so it's not divisible by 5.  However, 95 is divisible by 5. 
* **Find the pair:** 5 x 19 = 95

**Therefore, the factor pairs of 95 are:**

* 1 and 95
* 5 and 19 

Expected answer: [(1, 95), (5, 19)]

Correct? No

Problem: Please list all the factor pairs of 36.

Gemini-1.5 Flash:

Here are the factor pairs of 36:

* 1 and 36
* 2 and 18
* 3 and 12
* 4 and 9
* 6 and 6 

Expected answer: [(1, 36), (2, 18), (3, 12), (4, 9), (6, 6)]

Correct? No

Problem: Compute all the factor pairs of the number 59.

Gemini-1.5 Flash:

A factor pair is a set of two numbers that multiply together to equal the given number. 

Since 59 is a prime number, it is only divisible by 1 and itself.

Therefore, the only factor pair of 59 is **(1, 59)**. 

Expected answer: [(1, 59)]

Correct? No

Problem: What are all the factor pairs of 3?

Gemini-1.5 Flash:

The factor pairs of 3 are:

* **1 and 3** 

This is because 1 x 3 = 3. 
### Answer: 1 and 3 

Expected answer: [(1, 3)]

Correct? Yes

Problem: Could you list all factor pairs of 51?

Gemini-1.5 Flash:

Here are the factor pairs of 51:

* **1 and 51** 
* **3 and 17** 

Let me know if you'd like to explore factor pairs of other numbers! 

Expected answer: [(1, 51), (3, 17)]

Correct? No

Problem: List all factor pairs of 50.

Gemini-1.5 Flash:

Here are the factor pairs of 50:

* 1 and 50
* 2 and 25
* 5 and 10 

Expected answer: [(1, 50), (2, 25), (5, 10)]

Correct? No

Problem: Write down all factor pairs of 47.

Gemini-1.5 Flash:

The number 47 is a prime number.  Prime numbers only have two factors: 1 and itself. 

Therefore, the factor pairs of 47 are:

* **1 and 47** 

Expected answer: [(1, 47)]

Correct? No

Problem: Write down all factor pairs of 51.

Gemini-1.5 Flash:

The factor pairs of 51 are:

* 1 and 51
* 3 and 17 

Expected answer: [(1, 51), (3, 17)]

Correct? No

Problem: List all factor pairs of 60.

Gemini-1.5 Flash:

Here are the factor pairs of 60:

* 1 and 60
* 2 and 30
* 3 and 20
* 4 and 15
* 5 and 12
* 6 and 10 

Expected answer: [(1, 60), (2, 30), (3, 20), (4, 15), (5, 12), (6, 10)]

Correct? No

Problem: List all factor pairs of 43.

Gemini-1.5 Flash:

The only factor pairs of 43 are:

* **1 and 43** 

This is because 43 is a prime number, meaning it is only divisible by 1 and itself. 

Expected answer: [(1, 43)]

Correct? No

Problem: List all factor pairs of 70.

Gemini-1.5 Flash:

Here are all the factor pairs of 70:

* **1 and 70** 
* **2 and 35**
* **5 and 14**
* **7 and 10** 

Expected answer: [(1, 70), (2, 35), (5, 14), (7, 10)]

Correct? No

Problem: Please list all the factors of the number 2.

Gemini-1.5 Flash:

The factors of a number are the numbers that divide evenly into it.

The factors of 2 are:

* **1** (2 divided by 1 equals 2)
* **2** (2 divided by 2 equals 1)

### Answer: 1, 2 

Expected answer: [(1, 2)]

Correct? Yes

Problem: List all the factor pairs of 49.

Gemini-1.5 Flash:

The factor pairs of 49 are:

* **1 and 49** 
* **7 and 7** 

Expected answer: [(1, 49), (7, 7)]

Correct? No

Problem: Identify all factor pairs of 17.

Gemini-1.5 Flash:

A factor pair is a set of two numbers that multiply to give the original number.  

The only factors of 17 are 1 and 17. This is because 17 is a prime number, meaning it is only divisible by 1 and itself.

### Answer: (1, 17) 

Expected answer: [(1, 17)]

Correct? Yes

Problem: Find all the factor pairs of 35.

Gemini-1.5 Flash:

Here's how to find all the factor pairs of 35:

**Understanding Factors:**

* Factors are numbers that divide evenly into another number. 
* Factor pairs are two numbers that, when multiplied together, equal the original number.

**Finding the Factor Pairs of 35:**

1. **Start with 1:** 1 is always a factor of any number. The pair for 1 is 35 (1 x 35 = 35).

2. **Check for divisibility by 2, 3, and so on:** 35 is not divisible by 2 or 3.

3. **Try 5:** 35 is divisible by 5. The pair for 5 is 7 (5 x 7 = 35).

4. **Continue until you repeat numbers:** You've already found all the factors: 1, 5, 7, and 35.

**Factor Pairs of 35:**

* 1 and 35
* 5 and 7 

### Answer: (1, 35) and (5, 7) 

Expected answer: [(1, 35), (5, 7)]

Correct? Yes

Problem: Identify all factor pairs of the number 91.

Gemini-1.5 Flash:

Here's how to find the factor pairs of 91:

* **Start with 1:**  1 is always a factor of any number.  Since 91 / 1 = 91, the first factor pair is (1, 91).
* **Check divisibility by small prime numbers:**  Try 2, 3, 5, 7, etc.  You'll find that 91 is divisible by 7 (91 / 7 = 13).
* **The corresponding factor:**  Since 91 / 7 = 13, another factor pair is (7, 13).

**Therefore, the factor pairs of 91 are:**

* (1, 91)
* (7, 13) 

Expected answer: [(1, 91), (7, 13)]

Correct? No

Problem: List all factor pairs of the number 97.

Gemini-1.5 Flash:

The number 97 is a prime number.  Prime numbers only have two factors: 1 and itself.

### Answer: 1, 97 

Expected answer: [(1, 97)]

Correct? Yes

Problem: What are all the factor pairs of 16?

Gemini-1.5 Flash:

Here are the factor pairs of 16:

* **1 and 16** (1 x 16 = 16)
* **2 and 8** (2 x 8 = 16)
* **4 and 4** (4 x 4 = 16) 

Expected answer: [(1, 16), (2, 8), (4, 4)]

Correct? No

Problem: Find all the factor pairs of the number 21.

Gemini-1.5 Flash:

Here are the factor pairs of 21:

* **1 and 21** (1 x 21 = 21)
* **3 and 7** (3 x 7 = 21)

### Answer: (1, 21), (3, 7) 

Expected answer: [(1, 21), (3, 7)]

Correct? Yes

Problem: Could you determine all the factor pairs of 72?

Gemini-1.5 Flash:

Here are all the factor pairs of 72:

* **1 and 72** 
* **2 and 36**
* **3 and 24**
* **4 and 18**
* **6 and 12**
* **8 and 9** 
 
Let me know if you'd like me to explain how to find factor pairs! 

Expected answer: [(1, 72), (2, 36), (3, 24), (4, 18), (6, 12), (8, 9)]

Correct? No

Problem: Determine all the factor pairs of 38.

Gemini-1.5 Flash:

Here's how to find all the factor pairs of 38:

* **Start with 1:**  1 is always a factor of any number.  Since 38 divided by 1 is 38, our first factor pair is (1, 38).
* **Check for divisibility by 2:** 38 is even, so 2 is a factor. 38 divided by 2 is 19, giving us the factor pair (2, 19).
* **Check for divisibility by other numbers:**  38 is not divisible by 3, 4, 5, 6, 7, 8, or 9. 
* **We've reached the square root:** Since we're checking numbers less than the square root of 38, we know we've found all the factor pairs. 

**Therefore, the factor pairs of 38 are:**

* (1, 38)
* (2, 19) 

Expected answer: [(1, 38), (2, 19)]

Correct? No

Problem: List all the factor pairs of 94.

Gemini-1.5 Flash:

Here's how to find the factor pairs of 94:

* **Start by dividing 94 by the smallest prime number, 2:** 94 / 2 = 47. Since both 2 and 47 are whole numbers, they are a factor pair.
* **Continue dividing by prime numbers, checking if the result is a whole number:**
    * 94 / 3 is not a whole number.
    * 94 / 5 is not a whole number.
    * 94 / 7 is not a whole number.
    * 94 / 11 is not a whole number.
    * 94 / 13 is not a whole number.
    * 94 / 17 is not a whole number.
    * 94 / 19 is not a whole number.
    * 94 / 23 is not a whole number.
    * 94 / 29 is not a whole number.
    * 94 / 31 is not a whole number.
    * 94 / 37 is not a whole number.
    * 94 / 41 is not a whole number.
    * 94 / 43 is not a whole number. 
* **We've reached 47, which is the square root of 94. Since we haven't found any other factors, we're done.**

**Therefore, the factor pairs of 94 are:**

* 1 and 94
* 2 and 47 

Expected answer: [(1, 94), (2, 47)]

Correct? No

Problem: List all factor pairs of 32.

Gemini-1.5 Flash:

Here are the factor pairs of 32:

* **1 and 32** 
* **2 and 16**
* **4 and 8** 

Expected answer: [(1, 32), (2, 16), (4, 8)]

Correct? No

Problem: Identify all factor pairs of the number 33.

Gemini-1.5 Flash:

Here are the factor pairs of 33:

* **1 and 33** (1 x 33 = 33)
* **3 and 11** (3 x 11 = 33) 

Expected answer: [(1, 33), (3, 11)]

Correct? No

Problem: Write down all the factor pairs of 59.

Gemini-1.5 Flash:

The number 59 is a prime number. Prime numbers only have two factors: 1 and the number itself. 

Therefore, the only factor pairs of 59 are:

* **1 and 59** 

Expected answer: [(1, 59)]

Correct? No

Problem: List all the factor pairs of the number 54.

Gemini-1.5 Flash:

Here are the factor pairs of 54:

* **1 and 54** 
* **2 and 27**
* **3 and 18**
* **6 and 9** 

Expected answer: [(1, 54), (2, 27), (3, 18), (6, 9)]

Correct? No

Problem: Determine all factor pairs of the number 85.

Gemini-1.5 Flash:

Here's how to find all the factor pairs of 85:

* **Start with 1:**  1 is always a factor of any number. Since 85 / 1 = 85, the first factor pair is (1, 85).
* **Check divisibility by 5:**  85 ends in a 5, so it's divisible by 5.  85 / 5 = 17, giving us the factor pair (5, 17). 

**Therefore, the factor pairs of 85 are:**

* (1, 85)
* (5, 17) 

Expected answer: [(1, 85), (5, 17)]

Correct? No

Problem: Identify all the factor pairs of the number 56.

Gemini-1.5 Flash:

Here are the factor pairs of 56:

* **1 and 56** 
* **2 and 28**
* **4 and 14**
* **7 and 8** 

Expected answer: [(1, 56), (2, 28), (4, 14), (7, 8)]

Correct? No

Problem: List all the factor pairs of the number 24.

Gemini-1.5 Flash:

Here are the factor pairs of 24:

* 1 and 24
* 2 and 12
* 3 and 8
* 4 and 6 

Expected answer: [(1, 24), (2, 12), (3, 8), (4, 6)]

Correct? No

Problem: What are all the factor pairs of the number 84?

Gemini-1.5 Flash:

Here's how to find all the factor pairs of 84:

1. **Start with 1:**  1 is a factor of every number, so (1, 84) is a factor pair.

2. **Check divisibility by 2:** 84 is even, so 2 is a factor. 84/2 = 42, so (2, 42) is a factor pair.

3. **Check divisibility by 3:** The sum of the digits of 84 (8 + 4 = 12) is divisible by 3, so 84 is divisible by 3. 84/3 = 28, so (3, 28) is a factor pair.

4. **Check divisibility by 4:** 84 is divisible by 4 because the last two digits (84) are divisible by 4. 84/4 = 21, so (4, 21) is a factor pair.

5. **Check divisibility by 6:** 84 is divisible by 2 and 3, so it's also divisible by 6. 84/6 = 14, so (6, 14) is a factor pair.

6. **Check divisibility by 7:**  84 is divisible by 7 (you can use long division to confirm). 84/7 = 12, so (7, 12) is a factor pair.

7. **Continue until you "meet in the middle":** Notice that the factors we've found are getting closer together.  Since 7 x 12 = 84, we've found all the factor pairs.

**Therefore, the factor pairs of 84 are:**

* (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? No

Problem: Identify all factor pairs of the number 36.

Gemini-1.5 Flash:

Here are the factor pairs of 36:

* **1 and 36** 
* **2 and 18**
* **3 and 12**
* **4 and 9**
* **6 and 6** 

Expected answer: [(1, 36), (2, 18), (3, 12), (4, 9), (6, 6)]

Correct? No

Problem: What are all the factor pairs of 85?

Gemini-1.5 Flash:

Here's how to find the factor pairs of 85:

* **Start with 1:**  1 is always a factor of any number. Its pair is the number itself (85). So our first pair is (1, 85).
* **Check divisibility by other numbers:** 85 is odd, so it's not divisible by 2.  Check if it's divisible by 3, 5, 7, etc.  You'll find that 85 is divisible by 5.
* **Find the pair:** 85 divided by 5 is 17.  So our second pair is (5, 17). 

**Therefore, the factor pairs of 85 are:**

* (1, 85)
* (5, 17) 

Expected answer: [(1, 85), (5, 17)]

Correct? No

Problem: Identify all factor pairs of the number 34.

Gemini-1.5 Flash:

Here are the factor pairs of 34:

* **1 and 34** 
* **2 and 17** 

Since 34 is an even number, we know it will be divisible by 2.  We also see that 34 is not divisible by 3 or 5, so we can skip checking those numbers.  We've found all the factor pairs. 

Expected answer: [(1, 34), (2, 17)]

Correct? No

Problem: Determine all factor pairs of 88.

Gemini-1.5 Flash:

Here's how to find all the factor pairs of 88:

**1. Start with 1 and the number itself:**

* 1 x 88 = 88

**2.  Work your way up, checking if the number is divisible by each number:**

* 2 x 44 = 88
* 4 x 22 = 88
* 8 x 11 = 88 

**3. You've found all the factor pairs when you reach a number you've already used:** Notice that we've already used 11, so we're done.

**Therefore, the factor pairs of 88 are:**

* 1 and 88
* 2 and 44
* 4 and 22
* 8 and 11 

Expected answer: [(1, 88), (2, 44), (4, 22), (8, 11)]

Correct? No

Problem: Write down all factor pairs for the number 83.

Gemini-1.5 Flash:

The number 83 is a prime number. Prime numbers only have two factors: 1 and the number itself. 

Therefore, the only factor pairs for 83 are:

* **1 and 83** 

Expected answer: [(1, 83)]

Correct? No

Problem: What are all the factor pairs of 60?

Gemini-1.5 Flash:

Here's how to find all the factor pairs of 60:

**Understanding Factors**

Factors are numbers that divide evenly into another number. A factor pair is two numbers that multiply together to give you the original number.

**Finding the Pairs**

1. **Start with 1:** 1 is a factor of every number.  Its pair is 60 (1 x 60 = 60).
2. **Check 2:** 2 is a factor of 60 (60/2 = 30), so its pair is 30 (2 x 30 = 60).
3. **Continue with other numbers:**  Keep going, checking if each number divides evenly into 60.  If it does, find its pair.
   * 3 is a factor of 60 (60/3 = 20) - pair is 20
   * 4 is a factor of 60 (60/4 = 15) - pair is 15
   * 5 is a factor of 60 (60/5 = 12) - pair is 12
   * 6 is a factor of 60 (60/6 = 10) - pair is 10

**All the Factor Pairs of 60:**

* 1 and 60
* 2 and 30
* 3 and 20
* 4 and 15
* 5 and 12
* 6 and 10 

Expected answer: [(1, 60), (2, 30), (3, 20), (4, 15), (5, 12), (6, 10)]

Correct? No

Problem: Write down all factor pairs of 58.

Gemini-1.5 Flash:

The factor pairs of 58 are:

* 1 and 58
* 2 and 29 

Expected answer: [(1, 58), (2, 29)]

Correct? No

Problem: Please list all of the factor pairs of the number 40.

Gemini-1.5 Flash:

Here are the factor pairs of 40:

* **1 and 40**
* **2 and 20**
* **4 and 10**
* **5 and 8** 

Expected answer: [(1, 40), (2, 20), (4, 10), (5, 8)]

Correct? No

Problem: Identify all the factor pairs of 91.

Gemini-1.5 Flash:

Here's how to find the factor pairs of 91:

* **Start with 1:**  1 is a factor of every number, and its pair is the number itself (91). So, one factor pair is (1, 91).
* **Check divisibility by small primes:**  
    * 91 is not divisible by 2 (it's odd).
    * 91 is not divisible by 3 (the sum of the digits, 1+9+1 = 11, is not divisible by 3).
    * 91 is divisible by 7 (91 / 7 = 13).
* **The pair:** Since 7 is a factor, its pair is 13. 

**Therefore, the factor pairs of 91 are (1, 91) and (7, 13).** 

Expected answer: [(1, 91), (7, 13)]

Correct? No

Problem: List all the factor pairs of 76.

Gemini-1.5 Flash:

Here are the factor pairs of 76:

* **1 and 76**
* **2 and 38**
* **4 and 19** 

Expected answer: [(1, 76), (2, 38), (4, 19)]

Correct? No

Problem: What are all the factor pairs of 13?

Gemini-1.5 Flash:

The only factor pairs of 13 are:

* **1 and 13** 

This is because 13 is a prime number, meaning it is only divisible by 1 and itself. 

Expected answer: [(1, 13)]

Correct? No

Problem: Find all factor pairs of 64.

Gemini-1.5 Flash:

Here's how to find all the factor pairs of 64:

**Understanding Factors**

Factors are numbers that divide evenly into another number.  

**Finding the Pairs**

1. **Start with 1:**  1 is a factor of every number. Since 64 / 1 = 64, our first factor pair is (1, 64).

2. **Move to the next number:**  Try 2. 64 / 2 = 32, so our next factor pair is (2, 32).

3. **Continue:** Keep trying numbers, dividing 64 by them.  If the result is a whole number, you've found another factor pair.

* 64 / 4 = 16  -> (4, 16)
* 64 / 8 = 8  -> (8, 8) 

**Important Note:** Notice that when we get to 8, the factors start repeating (we already have 8 and 8). This is a signal that we've found all the factor pairs.

**List of Factor Pairs of 64**

* (1, 64)
* (2, 32)
* (4, 16)
* (8, 8) 

Expected answer: [(1, 64), (2, 32), (4, 16), (8, 8)]

Correct? No

Problem: What are all factor pairs of 8?

Gemini-1.5 Flash:

Here are the factor pairs of 8:

* **1 and 8** (1 x 8 = 8)
* **2 and 4** (2 x 4 = 8) 

Expected answer: [(1, 8), (2, 4)]

Correct? No

Problem: List all the factor pairs of 80.

Gemini-1.5 Flash:

Here are the factor pairs of 80:

* **1 and 80**
* **2 and 40**
* **4 and 20**
* **5 and 16**
* **8 and 10** 

Expected answer: [(1, 80), (2, 40), (4, 20), (5, 16), (8, 10)]

Correct? No

Problem: Calculate all the factor pairs for the number 96.

Gemini-1.5 Flash:

Here's how to find all the factor pairs for 96:

* **Start with 1:**  1 is a factor of every number, so the first pair is (1, 96).
* **Check divisibility by 2:** 96 is even, so 2 is a factor.  96/2 = 48, so the next pair is (2, 48).
* **Continue with other numbers:** Keep checking numbers to see if they divide evenly into 96.
    * 3 divides evenly into 96 (96/3 = 32), so we have (3, 32).
    * 4 divides evenly into 96 (96/4 = 24), so we have (4, 24).
    * 6 divides evenly into 96 (96/6 = 16), so we have (6, 16).
    * 8 divides evenly into 96 (96/8 = 12), so we have (8, 12).

Notice that as we keep checking, the factors we're finding are getting closer together. We've now reached a point where the next number we would check (9) is larger than the factors we've already found.  This means we've found all the factor pairs.

**Here are all the factor pairs of 96:**

* (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? No

Problem: Calculate all factor pairs of 19.

Gemini-1.5 Flash:

The only factor pairs of 19 are 1 and 19. This is because 19 is a prime number, meaning it is only divisible by 1 and itself.

### Answer: 1 and 19. 

Expected answer: [(1, 19)]

Correct? Yes

Problem: What are all factor pairs of 37?

Gemini-1.5 Flash:

A factor pair is two numbers that multiply together to equal a given number.  

The only factors of 37 are 1 and 37.  This is because 37 is a prime number, meaning it is only divisible by 1 and itself.

### Answer: 1 and 37. 

Expected answer: [(1, 37)]

Correct? Yes

Problem: Write down all factor pairs of the number 6.

Gemini-1.5 Flash:

Here are the factor pairs of 6:

* **1 and 6** (1 x 6 = 6)
* **2 and 3** (2 x 3 = 6) 

Expected answer: [(1, 6), (2, 3)]

Correct? No

Problem: List down all the factor pairs of 58.

Gemini-1.5 Flash:

Here are the factor pairs of 58:

* **1 and 58** 
* **2 and 29** 

That's it! 58 is only divisible by these four numbers. 

Expected answer: [(1, 58), (2, 29)]

Correct? No

Problem: List all the factor pairs of the number 81.

Gemini-1.5 Flash:

Here are the factor pairs of 81:

* **1 and 81** 
* **3 and 27**
* **9 and 9** 

Remember, a factor pair is two numbers that multiply together to give you the original number (in this case, 81). 

Expected answer: [(1, 81), (3, 27), (9, 9)]

Correct? No

Problem: Please list all factor pairs of the number 85.

Gemini-1.5 Flash:

Here are the factor pairs of 85:

* **1 and 85** 
* **5 and 17** 

These are all the possible combinations of two whole numbers that multiply together to equal 85. 

Expected answer: [(1, 85), (5, 17)]

Correct? No

Problem: Write down all factor pairs for the number 93.

Gemini-1.5 Flash:

Here are the factor pairs for 93:

* 1 and 93
* 3 and 31 

Expected answer: [(1, 93), (3, 31)]

Correct? No

Problem: Identify all factor pairs of 24.

Gemini-1.5 Flash:

Here are all the factor pairs of 24:

* **1 and 24** 
* **2 and 12**
* **3 and 8**
* **4 and 6** 

Expected answer: [(1, 24), (2, 12), (3, 8), (4, 6)]

Correct? No

Problem: List all the factor pairs of 97.

Gemini-1.5 Flash:

The only factor pairs of 97 are:

* **1 and 97** 

This is because 97 is a prime number, meaning it is only divisible by 1 and itself. 

Expected answer: [(1, 97)]

Correct? No

Problem: Identify all factor pairs of the number 57.

Gemini-1.5 Flash:

A factor pair of a number is two numbers that multiply together to equal that number. 

Let's find the factor pairs of 57:

* **1 and 57:** 1 * 57 = 57
* **3 and 19:** 3 * 19 = 57

Therefore, the factor pairs of 57 are **(1, 57)** and **(3, 19)**. 

Expected answer: [(1, 57), (3, 19)]

Correct? No

Problem: List down all the factor pairs of 19.

Gemini-1.5 Flash:

The only factor pairs of 19 are:

* **1 and 19** 

This is because 19 is a prime number, meaning it is only divisible by 1 and itself. 

Expected answer: [(1, 19)]

Correct? No

Problem: What are all factor pairs of 38?

Gemini-1.5 Flash:

Here's how to find the factor pairs of 38:

* **Start with 1:**  1 is always a factor of any number.  Since 38 / 1 = 38, the first factor pair is (1, 38).
* **Check divisibility by 2:**  38 is even, so it's divisible by 2.  38 / 2 = 19, giving us the factor pair (2, 19). 
* **Continue checking:** Since 3 is not a factor of 38, and 4 is larger than 2, you've found all the factor pairs.

**Therefore, the factor pairs of 38 are:**

* (1, 38)
* (2, 19) 

Expected answer: [(1, 38), (2, 19)]

Correct? No

Problem: Find all the factor pairs of the number 6.

Gemini-1.5 Flash:

A factor pair is two numbers that multiply together to give you the original number. 

Here are the factor pairs of 6:

* 1 and 6 (1 x 6 = 6)
* 2 and 3 (2 x 3 = 6) 

### Answer: (1, 6) and (2, 3) 

Expected answer: [(1, 6), (2, 3)]

Correct? Yes

Problem: Write down all factor pairs of 17.

Gemini-1.5 Flash:

The only factor pairs of 17 are:

* **1 and 17** 

This is because 17 is a prime number, meaning it is only divisible by 1 and itself. 

Expected answer: [(1, 17)]

Correct? No

Problem: List all factor pairs of 45.

Gemini-1.5 Flash:

Here are the factor pairs of 45:

* **1 and 45** 
* **3 and 15**
* **5 and 9** 

Expected answer: [(1, 45), (3, 15), (5, 9)]

Correct? No

Problem: Write down all factor pairs of 36.

Gemini-1.5 Flash:

Here are all the factor pairs of 36:

* **1 and 36**
* **2 and 18**
* **3 and 12**
* **4 and 9**
* **6 and 6** 

Expected answer: [(1, 36), (2, 18), (3, 12), (4, 9), (6, 6)]

Correct? No

Problem: What are all factor pairs of 89?

Gemini-1.5 Flash:

The only factor pairs of 89 are 1 and 89. This is because 89 is a prime number, meaning it is only divisible by 1 and itself. 

### Answer: 1 and 89 

Expected answer: [(1, 89)]

Correct? Yes

Problem: List all the factor pairs of 89.

Gemini-1.5 Flash:

The number 89 is a prime number. Prime numbers only have two factors: 1 and itself.

Therefore, the factor pairs of 89 are:

* **1 and 89** 

Expected answer: [(1, 89)]

Correct? No

Problem: List all the factor pairs of the number 44.

Gemini-1.5 Flash:

Here are the factor pairs of 44:

* **1 and 44** 
* **2 and 22**
* **4 and 11** 

Expected answer: [(1, 44), (2, 22), (4, 11)]

Correct? No

Problem: Identify all factor pairs of the number 36.

Gemini-1.5 Flash:

Here are the factor pairs of 36:

* **1 and 36** (1 x 36 = 36)
* **2 and 18** (2 x 18 = 36)
* **3 and 12** (3 x 12 = 36)
* **4 and 9** (4 x 9 = 36)
* **6 and 6** (6 x 6 = 36) 

Expected answer: [(1, 36), (2, 18), (3, 12), (4, 9), (6, 6)]

Correct? No

Problem: List all possible factor pairs of the number 35.

Gemini-1.5 Flash:

Here are the factor pairs of 35:

* **1 and 35** 
* **5 and 7** 

Expected answer: [(1, 35), (5, 7)]

Correct? No

Problem: Write down all factor pairs of 22.

Gemini-1.5 Flash:

Here are the factor pairs of 22:

* **1 and 22** 
* **2 and 11** 

Expected answer: [(1, 22), (2, 11)]

Correct? No

Problem: Find all the factor pairs of 40.

Gemini-1.5 Flash:

Here's how to find all the factor pairs of 40:

**Understanding Factors**

Factors are numbers that divide evenly into another number.  A factor pair is a set of two factors that multiply together to give you the original number.

**Finding the Factor Pairs**

1. **Start with 1:**  1 is always a factor of any number.  The pair with 1 is 40 (1 x 40 = 40).

2. **Check divisibility by 2:**  40 is even, so 2 is a factor.  The pair with 2 is 20 (2 x 20 = 40).

3. **Check divisibility by 4:** 40 is divisible by 4. The pair with 4 is 10 (4 x 10 = 40).

4. **Check divisibility by 5:** 40 ends in a 0, so it's divisible by 5. The pair with 5 is 8 (5 x 8 = 40).

5. **Check divisibility by 8:** We already found 8 as a factor.

6. **Check divisibility by 10:** We already found 10 as a factor.

**The Factor Pairs of 40 are:**

* 1 and 40
* 2 and 20
* 4 and 10
* 5 and 8 

Expected answer: [(1, 40), (2, 20), (4, 10), (5, 8)]

Correct? No

Problem: List all the factor pairs of the number 64.

Gemini-1.5 Flash:

Here are all the factor pairs of 64:

* **1 and 64**
* **2 and 32**
* **4 and 16**
* **8 and 8** 

Expected answer: [(1, 64), (2, 32), (4, 16), (8, 8)]

Correct? No

Problem: Determine all factor pairs of 30.

Gemini-1.5 Flash:

Here's how to find all the factor pairs of 30:

**Understanding Factors**

Factors are numbers that divide evenly into another number.  

**Finding the Pairs**

1. **Start with 1:** 1 is a factor of every number, and its pair is the number itself (30).  So our first pair is (1, 30).

2. **Check divisibility by 2:**  30 is even, so 2 is a factor. 30 / 2 = 15, so our next pair is (2, 15).

3. **Check divisibility by 3:** 30 is divisible by 3. 30 / 3 = 10, giving us the pair (3, 10).

4. **Check divisibility by 4:** 30 is not divisible by 4.

5. **Check divisibility by 5:** 30 is divisible by 5. 30 / 5 = 6, giving us the pair (5, 6).

**All the Factor Pairs of 30:**

* (1, 30)
* (2, 15)
* (3, 10)
* (5, 6) 

Expected answer: [(1, 30), (2, 15), (3, 10), (5, 6)]

Correct? No

Problem: List all factor pairs of 59.

Gemini-1.5 Flash:

The only factor pairs of 59 are:

* **1 and 59** 

59 is a prime number, meaning it is only divisible by 1 and itself. 

Expected answer: [(1, 59)]

Correct? No

Problem: List all factor pairs of 18.

Gemini-1.5 Flash:

Here are the factor pairs of 18:

* **1 and 18** 
* **2 and 9**
* **3 and 6** 

Expected answer: [(1, 18), (2, 9), (3, 6)]

Correct? No

Problem: Please list all the factor pairs of 61.

Gemini-1.5 Flash:

The only factor pairs of 61 are:

* **1 and 61** 

This is because 61 is a prime number, meaning it is only divisible by 1 and itself. 

Expected answer: [(1, 61)]

Correct? No

Problem: What are all of the factor pairs of 86?

Gemini-1.5 Flash:

Here's how to find the factor pairs of 86:

* **Start by trying to divide 86 by 1:** 86 / 1 = 86, so (1, 86) is a factor pair.
* **Continue dividing 86 by increasing whole numbers.**  You'll find that 86 is only divisible by 1, 2, 43, and 86.

**Therefore, the factor pairs of 86 are:**

* (1, 86)
* (2, 43) 

Expected answer: [(1, 86), (2, 43)]

Correct? No

Problem: List out all factor pairs of 57.

Gemini-1.5 Flash:

The factor pairs of 57 are:

* 1 and 57
* 3 and 19 

Expected answer: [(1, 57), (3, 19)]

Correct? No

Problem: What are all factor pairs of 38?

Gemini-1.5 Flash:

Here's how to find the factor pairs of 38:

**Understanding Factors**

Factors are numbers that divide evenly into another number.  A factor pair is a set of two numbers that multiply to give you the original number.

**Finding the Factor Pairs**

1. **Start with 1:**  1 is always a factor of any number. Its pair is the number itself: 1 x 38 = 38.

2. **Check divisibility by 2:**  38 is even, so 2 is a factor.  Its pair is 19: 2 x 19 = 38.

**The Factor Pairs of 38:**

* 1 and 38
* 2 and 19 

**Important Note:**  We've found all the factor pairs because 19 is a prime number (only divisible by 1 and itself). 

Expected answer: [(1, 38), (2, 19)]

Correct? No

Problem: Please list all factor pairs of 27.

Gemini-1.5 Flash:

Here are the factor pairs of 27:

* **1 and 27** 
* **3 and 9** 

These are all the pairs of whole numbers that multiply together to equal 27. 

Expected answer: [(1, 27), (3, 9)]

Correct? No

Problem: List all the factor pairs of the number 78.

Gemini-1.5 Flash:

Here are the factor pairs of 78:

* **1 and 78**
* **2 and 39**
* **3 and 26**
* **6 and 13** 

Expected answer: [(1, 78), (2, 39), (3, 26), (6, 13)]

Correct? No

Problem: List out all the factor pairs of 23.

Gemini-1.5 Flash:

The only factor pairs of 23 are:

* **1 and 23** 

This is because 23 is a prime number, meaning it is only divisible by 1 and itself. 

Expected answer: [(1, 23)]

Correct? No

Problem: List down all the factor pairs of 2.

Gemini-1.5 Flash:

The only factor pairs of 2 are:

* **1 and 2** 
 
This is because 1 x 2 = 2. 

Expected answer: [(1, 2)]

Correct? No

Problem: Write down all the factor pairs of 49.

Gemini-1.5 Flash:

The factor pairs of 49 are:

* 1 and 49
* 7 and 7 

Expected answer: [(1, 49), (7, 7)]

Correct? No

Problem: Identify all the distinct pairs of factors that can multiply to give the number 93.

Gemini-1.5 Flash:

Here's how to find the factors of 93:

1. **Start with 1 and the number itself:**  1 and 93 are always factors.

2. **Check divisibility by small primes:**  
   * 93 is not divisible by 2 (it's odd).
   * 93 is not divisible by 3 (the sum of its digits, 9+3=12, is not divisible by 3).
   * 93 is divisible by 3, since 93 / 3 = 31.

3. **Check if the new factor has a pair:** Since 31 is a prime number, its only factors are 1 and itself.

Therefore, the distinct pairs of factors of 93 are:

* **1 and 93**
* **3 and 31** 

Expected answer: [(1, 93), (3, 31)]

Correct? No