🔗 Problem

Problem: Find all factor pairs of 64.

Claude-3 Sonnet:

    To find all factor pairs of a number, we need to find all the pairs of numbers that multiply together to give that number.

    For 64, we can start by finding the factors:
    1 x 64 = 64
    2 x 32 = 64
    4 x 16 = 64
    8 x 8 = 64

    These are all the factor pairs of 64.

    We can list them out as follows:
    (1, 64)
    (2, 32)
    (4, 16)
    (8, 8)

    Note that (8, 8) is the same as (8, 8), so we only need to list it once.

    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

Problem: List all the factor pairs of 18.

Claude-3 Sonnet:

    To find the factor pairs of a number, we need to find all the pairs of numbers that multiply together to give that number.

    The factors of 18 are: 1, 2, 3, 6, 9, 18.

    The factor pairs of 18 are:
    1 and 18
    2 and 9
    3 and 6

    Therefore, the list of all factor pairs of 18 is:
    (1, 18)
    (2, 9)
    (3, 6)

    ### Answer: (1, 18), (2, 9), (3, 6)

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

Correct? Yes

🔗 Problem

Problem: Find all factor pairs of 28.

Claude-3 Sonnet:

    To find all factor pairs of a number, we need to find all the pairs of numbers that multiply together to give that number.

    Here are the steps to find all factor pairs of 28:

    1. List out all the factors of 28.
       The factors of 28 are: 1, 2, 4, 7, 14, 28

    2. Pair up each factor with its corresponding factor pair.
       1 and 28 (1 × 28 = 28)
       2 and 14 (2 × 14 = 28)
       4 and 7 (4 × 7 = 28)

    Therefore, the factor pairs of 28 are:
    (1, 28)
    (2, 14)
    (4, 7)

    ### Answer: The factor pairs of 28 are (1, 28), (2, 14), and (4, 7).

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

Correct? Yes

🔗 Problem

Problem: List all factor pairs of the number 48.

Claude-3 Sonnet:

    To find the factor pairs of a number, we need to find all the pairs of numbers that multiply together to give that number.

    The factor pairs of 48 are:

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

    These are all the factor pairs because:
    1 × 48 = 48
    2 × 24 = 48
    3 × 16 = 48
    4 × 12 = 48
    6 × 8 = 48

    Therefore, the list of all factor pairs of 48 is:
    (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

Problem: Write down all factor pairs of 7.

Claude-3 Sonnet:

    To find the factor pairs of a number, we need to find all the pairs of numbers that multiply together to give that number.

    The factor pairs of 7 are:

    1 and 7
    7 and 1

    Since 7 is a prime number, it has no other factors besides 1 and itself.

    Therefore, the factor pairs of 7 are:
    1, 7
    7, 1

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

Expected answer: [(1, 7)]

Correct? Yes

🔗 Problem

Problem: Write down every pair of factors of 98.

Claude-3 Sonnet:

    To find all the pairs of factors of 98, we need to find the numbers that divide into 98 evenly, and pair each factor with its corresponding factor.

    The factors of 98 are:
    1, 2, 7, 14, 49, 98

    The pairs of factors are:
    1 and 98
    2 and 49
    7 and 14

    Therefore, the pairs of factors of 98 are:
    1 and 98
    2 and 49
    7 and 14
    14 and 7
    49 and 2
    98 and 1

    ### Answer: 1 and 98, 2 and 49, 7 and 14, 14 and 7, 49 and 2, 98 and 1

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

Correct? Yes

🔗 Problem

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

Claude-3 Sonnet:

    To find the factor pairs of a number, we need to find all the pairs of numbers that multiply together to give that number.

    The factors of 26 are:
    1, 26
    2, 13

    These are the only factor pairs of 26 because 26 is not divisible by any other numbers besides 1, 2, 13, and 26.

    Therefore, the list of all factor pairs of 26 is:
    1, 26
    2, 13

    ### Answer: (1, 26), (2, 13)

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

Correct? Yes

🔗 Problem

Problem: Write down all factor pairs of 59.

Claude-3 Sonnet:

    To find all the factor pairs of 59, we need to find all the numbers that divide into 59 evenly, along with their corresponding pairs.

    The factors of 59 are:
    1 and 59
    59 and 1

    To find the remaining factor pairs, we can divide 59 by each number from 2 to the square root of 59 (approximately 7.68) and check if the division is exact. If it is, then we have found a factor pair.

    2 goes into 59 with a remainder, so 2 is not a factor.
    3 goes into 59 with a remainder, so 3 is not a factor.
    4 goes into 59 with a remainder, so 4 is not a factor.
    5 goes into 59 with a remainder, so 5 is not a factor.
    6 goes into 59 with a remainder, so 6 is not a factor.
    7 goes into 59 with a remainder, so 7 is not a factor.

    Therefore, the only factor pairs of 59 are:
    1 and 59
    59 and 1

    ### Answer: The factor pairs of 59 are (1, 59) and (59, 1).

Expected answer: [(1, 59)]

Correct? Yes

🔗 Problem

Problem: Find out all factor pairs of 20.

Claude-3 Sonnet:

    To find all factor pairs of a number, we need to find all the pairs of numbers that multiply together to give that number.

    For the number 20, the factor pairs are:

    1 and 20
    2 and 10
    4 and 5

    Explanation:
    1 × 20 = 20
    2 × 10 = 20
    4 × 5 = 20

    These are all the factor pairs of 20.

    ### 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

Problem: Write down all factor pairs of 20.

Claude-3 Sonnet:

    To find the factor pairs of 20, we need to find all the pairs of numbers that multiply together to give 20.

    The factor pairs of 20 are:

    1 and 20
    2 and 10
    4 and 5

    These are all the factor pairs because:
    1 × 20 = 20
    2 × 10 = 20
    4 × 5 = 20

    There are no other pairs of numbers that multiply together to give 20.

    ### 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

Problem: Please list down all factor pairs of 13.

Claude-3 Sonnet:

    To find the factor pairs of a number, we need to find all the pairs of numbers that multiply together to give that number.

    The factor pairs of 13 are:

    1 and 13
    13 and 1

    Since 13 is a prime number, it has no other factors besides 1 and itself.

    Therefore, the complete list of factor pairs of 13 is:

    1, 13
    13, 1

    ### Answer: (1, 13), (13, 1)

Expected answer: [(1, 13)]

Correct? Yes

🔗 Problem

Problem: Find all factor pairs of 81.

Claude-3 Sonnet:

    To find all factor pairs of a number, we need to find all the pairs of numbers that multiply together to give that number.

    For 81, we can follow these steps:

    1. Start with 1 and 81 as the first pair of factors.
    2. Divide 81 by 2 to get 40.5. Since 81 is not divisible by 2, we move on.
    3. Divide 81 by 3 to get 27. This is a factor pair: 3 and 27.
    4. Divide 81 by 4 to get 20.25. Since 81 is not divisible by 4, we move on.
    5. Divide 81 by 5 to get 16.2. Since 81 is not divisible by 5, we move on.
    6. Divide 81 by 6 to get 13.5. Since 81 is not divisible by 6, we move on.
    7. Divide 81 by 7 to get 11.57. Since 81 is not divisible by 7, we move on.
    8. Divide 81 by 8 to get 10.125. Since 81 is not divisible by 8, we move on.
    9. Divide 81 by 9 to get 9. This is a factor pair: 9 and 9.

    Therefore, the factor pairs of 81 are:
    1 and 81
    3 and 27
    9 and 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

Problem: List all factor pairs of 76.

Claude-3 Sonnet:

    To find the factor pairs of 76, we need to find all the pairs of numbers that multiply together to give 76.

    The factors of 76 are:
    1, 76
    2, 38
    4, 19

    Therefore, the factor pairs of 76 are:
    1 and 76
    2 and 38
    4 and 19

    ### Answer: (1, 76), (2, 38), (4, 19)

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

Correct? Yes

🔗 Problem

Problem: Write down all factor pairs of 50.

Claude-3 Sonnet:

    To find the factor pairs of 50, we need to find all the pairs of numbers that multiply together to give 50.

    The factor pairs of 50 are:

    1 and 50
    2 and 25
    5 and 10

    These are all the factor pairs because:
    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

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

Claude-3 Sonnet:

    To find the factor pairs of a number, we need to find all the pairs of numbers that multiply together to give that number.

    Here are the steps to find the factor pairs of 76:

    1. Start with 1 and 76 as the first pair.
    2. Find the next pair by dividing 76 by the next number greater than 1, which is 2.
       76 ÷ 2 = 38, so the next pair is 2 and 38.
    3. Continue dividing 76 by the next greater number until the quotient becomes smaller than the divisor.

    The factor pairs of 76 are:
    1 and 76
    2 and 38
    4 and 19

    Therefore, the list of all factor pairs of 76 is:
    (1, 76), (2, 38), (4, 19)

    ### Answer: (1, 76), (2, 38), (4, 19)

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

Correct? Yes

🔗 Problem

Problem: List all factor pairs of 56.

Claude-3 Sonnet:

    To find the factor pairs of 56, we need to find all the pairs of numbers that multiply together to give 56.

    The factors of 56 are:
    1, 56
    2, 28
    4, 14
    7, 8

    Therefore, the factor pairs of 56 are:
    1 and 56
    2 and 28
    4 and 14
    7 and 8

    ### Answer: (1, 56), (2, 28), (4, 14), (7, 8)

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

Correct? Yes

🔗 Problem

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

Claude-3 Sonnet:

    To find the factor pairs of a number, we need to find all the pairs of numbers that multiply together to give that number.

    The factors of 44 are:
    1, 44
    2, 22
    4, 11

    Therefore, the factor pairs of 44 are:
    1 and 44
    2 and 22
    4 and 11

    ### 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

Problem: List all the factor pairs of 5.

Claude-3 Sonnet:

    To find the factor pairs of a number, we need to find all the pairs of numbers that multiply together to give that number.

    The factor pairs of 5 are:

    1 and 5

    Since 5 is a prime number, it has no other factor pairs besides 1 and itself.

    Therefore, the list of all factor pairs of 5 is:

    (1, 5)

    ### Answer: (1, 5)

Expected answer: [(1, 5)]

Correct? Yes

🔗 Problem

Problem: Identify all factor pairs of the number 95.

Claude-3 Sonnet:

    To find all factor pairs of a number, we need to find all the pairs of numbers that multiply together to give that number.

    Here are the steps to find the factor pairs of 95:

    1. List out all the factors of 95.
       The factors of 95 are: 1, 5, 19, 95

    2. Pair up each factor with its corresponding factor pair.
       1 and 95 (1 × 95 = 95)
       5 and 19 (5 × 19 = 95)

    Therefore, the factor pairs of 95 are:
    1 and 95
    5 and 19

    ### Answer: The factor pairs of 95 are 1 and 95, 5 and 19.

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

Correct? Yes

🔗 Problem

Problem: Please list all the factor pairs of 36.

Claude-3 Sonnet:

    To find the factor pairs of 36, we need to find all the pairs of numbers that multiply together to give 36.

    The factor pairs of 36 are:

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

    Note that 6 and 6 is the same pair, so we only list it once.

    ### 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

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

Claude-3 Sonnet:

    To find all the factor pairs of a number, we need to find all the pairs of numbers that multiply together to give that number.

    Here are the steps to find the factor pairs of 59:

    1. Start with 1 and the number itself (59).
    2. Check if 1 divides 59 evenly. 59 ÷ 1 = 59, so 1 and 59 are a factor pair.
    3. Check if 2 divides 59 evenly. 59 ÷ 2 is not a whole number, so 2 is not a factor of 59.
    4. Check if 3 divides 59 evenly. 59 ÷ 3 is not a whole number, so 3 is not a factor of 59.
    5. Check if 4 divides 59 evenly. 59 ÷ 4 is not a whole number, so 4 is not a factor of 59.
    6. Check if 5 divides 59 evenly. 59 ÷ 5 is not a whole number, so 5 is not a factor of 59.
    7. Check if 6 divides 59 evenly. 59 ÷ 6 is not a whole number, so 6 is not a factor of 59.
    8. Check if 7 divides 59 evenly. 59 ÷ 7 is not a whole number, so 7 is not a factor of 59.
    9. Check if 8 divides 59 evenly. 59 ÷ 8 is not a whole number, so 8 is not a factor of 59.
    10. Check if 9 divides 59 evenly. 59 ÷ 9 is not a whole number, so 9 is not a factor of 59.
    11. Check if 10 divides 59 evenly. 59 ÷ 10 is not a whole number, so 10 is not a factor of 59.
    12. Check if 11 divides 59 evenly. 59 ÷ 11 is not a whole number, so 11 is not a factor of 59.
    13. Check if 12 divides 59 evenly. 59 ÷ 12 is not a whole number, so 12 is not a factor of 59.
    14. Check if 13 divides 59 evenly. 59 ÷ 13 = 4 (with a remainder of 7), so 13 is not a factor of 59.
    15. Check if 14 divides 59 evenly. 59 ÷ 14 is not a whole number, so 14 is not a factor of 59.
    16. Check if 15 divides 59 evenly. 59 ÷ 15 is not a whole number, so 15 is not a factor of 59.
    17. Check if 16 divides 59 evenly. 59 ÷ 16 is not a whole number, so 16 is not a factor of 59.
    18. Check if 17 divides 59 evenly. 59 ÷ 17 = 3 (with a remainder of 16), so 17 and 3 are a factor pair.
    19. Check if 18 divides 59 evenly. 59 ÷ 18 is not a whole number, so 18 is not a factor of 59.
    20. Check if 19 divides 59 evenly. 59 ÷ 19 = 3 (with a remainder of 2), so 19 is not a factor of 59.
    21. Check if 20 divides 59 evenly. 59 ÷ 20 is not a whole number, so 20 is not a factor of 59.
    22. Check if 21 divides 59 evenly. 59 ÷ 21 is not a whole number, so 21 is not a factor of 59.
    23

Expected answer: [(1, 59)]

Correct? No

🔗 Problem

Problem: What are all the factor pairs of 3?

Claude-3 Sonnet:

    To find the factor pairs of a number, we need to find all the pairs of numbers that multiply together to give that number.

    For the number 3, the factor pairs are:

    1 and 3

    Since 3 is a prime number, it only has one factor pair: 1 and itself.

    Therefore, the factor pairs of 3 are:
    (1, 3)

    Note that the order of the factors in the pair does not matter, so (1, 3) and (3, 1) represent the same factor pair.

    ### Answer: (1, 3)

Expected answer: [(1, 3)]

Correct? Yes

🔗 Problem

Problem: Could you list all factor pairs of 51?

Claude-3 Sonnet:

    To find the factor pairs of 51, we need to find all the numbers that divide evenly into 51.

    The factors of 51 are:
    1 and 51
    3 and 17

    Therefore, the factor pairs of 51 are:
    1, 51
    3, 17

    ### Answer: (1, 51), (3, 17)

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

Correct? Yes

🔗 Problem

Problem: List all factor pairs of 50.

Claude-3 Sonnet:

    To find the factor pairs of 50, we need to find all the pairs of numbers that multiply together to give 50.

    The factor pairs of 50 are:

    1 and 50
    2 and 25
    5 and 10

    These are all the factor pairs because 50 is not divisible by any other numbers besides 1, 2, 5, 10, 25, and 50.

    ### Answer: (1, 50), (2, 25), (5, 10)

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

Correct? Yes

🔗 Problem

Problem: Write down all factor pairs of 47.

Claude-3 Sonnet:

    To find the factor pairs of 47, we need to find all the numbers that divide evenly into 47.

    The factors of 47 are:
    1 and 47 (47 ÷ 1 = 47, 47 ÷ 47 = 1)

    To find the remaining factor pairs, we can divide 47 by each number from 2 to the square root of 47 (approximately 6.86) and check if the division is exact (with no remainder). If it is, then we have found a factor pair.

    2 does not divide evenly into 47 (47 ÷ 2 = 23.5, with a remainder of 1)
    3 does not divide evenly into 47 (47 ÷ 3 = 15.67, with a remainder of 2)
    4 does not divide evenly into 47 (47 ÷ 4 = 11.75, with a remainder of 3)
    5 does not divide evenly into 47 (47 ÷ 5 = 9.4, with a remainder of 2)
    6 does not divide evenly into 47 (47 ÷ 6 = 7.83, with a remainder of 5)
    7 divides evenly into 47 (47 ÷ 7 = 6.71, with no remainder)

    Therefore, the factor pairs of 47 are:
    1 and 47
    7 and (47 ÷ 7 = 6.71)

    ### Answer: The factor pairs of 47 are 1 and 47, 7 and 6.71.

Expected answer: [(1, 47)]

Correct? No

🔗 Problem

Problem: Write down all factor pairs of 51.

Claude-3 Sonnet:

    To find the factor pairs of 51, we need to find all the numbers that divide evenly into 51.

    The factors of 51 are:
    1 and 51
    3 and 17

    Therefore, the factor pairs of 51 are:
    1, 51
    3, 17

    ### Answer: (1, 51), (3, 17)

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

Correct? Yes

🔗 Problem

Problem: List all factor pairs of 60.

Claude-3 Sonnet:

    To find the factor pairs of 60, we need to find all the pairs of numbers that multiply together to give 60.

    The factors of 60 are: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60.

    The factor pairs of 60 are:

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

    Note that each pair is listed in both orders, for example, 2 and 30, as well as 30 and 2.

    ### 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

Problem: List all factor pairs of 43.

Claude-3 Sonnet:

    To find the factor pairs of 43, we need to find all the pairs of numbers that multiply together to give 43.

    The factors of 43 are:
    1, 43
    43, 1

    To find the remaining factor pairs, we can divide 43 by each number from 2 up to the square root of 43 (which is approximately 6.56) and check if the division is exact (i.e., the remainder is zero). If so, then we have found a factor pair.

    2 goes into 43 with a remainder, so 2 is not a factor.
    3 goes into 43 with a remainder, so 3 is not a factor.
    4 goes into 43, leaving a remainder of 3, so 4 is not a factor.
    5 goes into 43, leaving a remainder of 3, so 5 is not a factor.
    6 goes into 43, leaving a remainder of 1, so 6 is not a factor.
    7 goes into 43, leaving a remainder of 0, so 7 is a factor. The factor pair is 7 and 43/7 = 6.

    Therefore, the factor pairs of 43 are:
    1, 43
    43, 1
    7, 6
    6, 7

    ### Answer: The factor pairs of 43 are: (1, 43), (43, 1), (7, 6), (6, 7).

Expected answer: [(1, 43)]

Correct? No

🔗 Problem

Problem: List all factor pairs of 70.

Claude-3 Sonnet:

    To find the factor pairs of 70, we need to find all the pairs of numbers that multiply together to give 70.

    The factors of 70 are:
    1, 70
    2, 35
    5, 14
    7, 10

    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

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

Claude-3 Sonnet:

    The factors of 2 are:

    1
    2

    These are the only two factors of 2 because 2 is a prime number.

    ### Answer: 1, 2

Expected answer: [(1, 2)]

Correct? Yes

🔗 Problem

Problem: List all the factor pairs of 49.

Claude-3 Sonnet:

    To find the factor pairs of a number, we need to find all the pairs of numbers that multiply together to give that number.

    The factors of 49 are:
    1 and 49
    7 and 7

    To list all the factor pairs, we need to include both orders of each pair:
    1, 49
    49, 1
    7, 7

    Therefore, the list of all factor pairs of 49 is:
    1, 49
    49, 1
    7, 7

    ### Answer: 1, 49; 49, 1; 7, 7

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

Correct? Yes

🔗 Problem

Problem: Identify all factor pairs of 17.

Claude-3 Sonnet:

    To find the factor pairs of 17, we need to find all the numbers that divide evenly into 17.

    The factors of 17 are:
    1 and 17 (since 1 × 17 = 17)

    There are no other factor pairs of 17 because it is a prime number.

    Therefore, the only factor pair of 17 is:
    1 and 17

    ### Answer: 1 and 17

Expected answer: [(1, 17)]

Correct? Yes

🔗 Problem

Problem: Find all the factor pairs of 35.

Claude-3 Sonnet:

    To find all the factor pairs of a number, we need to find all the pairs of numbers that multiply together to give that number.

    Here's how we can find the factor pairs of 35:

    1) First, we list out all the factors of 35:
       Factors of 35 = 1, 5, 7, 35

    2) Next, we pair up the factors such that their product is 35:
       1 x 35
       5 x 7

    Therefore, the factor pairs of 35 are:
    (1, 35) and (5, 7)

    Note: The order of the factors in each pair doesn't matter, so (5, 7) is the same as (7, 5).

    ### Answer: The factor pairs of 35 are (1, 35) and (5, 7).

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

Correct? Yes

🔗 Problem

Problem: Identify all factor pairs of the number 91.

Claude-3 Sonnet:

    To find all factor pairs of a number, we need to find all the pairs of numbers that multiply together to give that number.

    Here are the steps to find the factor pairs of 91:

    1. Start with 1 and the number itself (91).
       This is always a factor pair: 1 × 91 = 91

    2. Check for other factors by dividing 91 by consecutive numbers starting from 2.
       If the division is exact (with no remainder), then the divisor and the quotient form a factor pair.

    3. Stop when the divisor becomes greater than the square root of 91 (approximately 9.54).

    Following these steps, we get the factor pairs of 91:

    1 × 91
    7 × 13

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

    ### Answer: The factor pairs of 91 are (1, 91) and (7, 13).

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

Correct? Yes

🔗 Problem

Problem: List all factor pairs of the number 97.

Claude-3 Sonnet:

    To find the factor pairs of a number, we need to find all the pairs of numbers that multiply together to give that number.

    Here are the steps to find the factor pairs of 97:

    1. Start with 1 and 97 as the first pair of factors.
    2. Check if any number between 1 and 97 divides 97 evenly.
    3. If a number divides 97 evenly, pair it with the quotient obtained by dividing 97 by that number.
    4. Repeat step 3 until you reach the square root of 97.

    Following these steps, we get the factor pairs of 97 as:

    1 and 97
    97 and 1

    Now, let's check for other factors between 1 and the square root of 97 (which is approximately 9.85).

    2 does not divide 97 evenly.
    3 does not divide 97 evenly.
    4 does not divide 97 evenly.
    5 does not divide 97 evenly.
    6 does not divide 97 evenly.
    7 does not divide 97 evenly.
    8 does not divide 97 evenly.
    9 does not divide 97 evenly.

    Since no other number between 1 and the square root of 97 divides 97 evenly, we have found all the factor pairs of 97.

    Therefore, the factor pairs of 97 are:
    1 and 97
    97 and 1

    ### Answer: The factor pairs of 97 are 1 and 97, 97 and 1.

Expected answer: [(1, 97)]

Correct? Yes

🔗 Problem

Problem: What are all the factor pairs of 16?

Claude-3 Sonnet:

    To find the factor pairs of a number, we need to find all the pairs of numbers that multiply together to give that number.

    The factors of 16 are:
    1, 16
    2, 8
    4, 4

    So, the factor pairs of 16 are:
    1 and 16
    2 and 8
    4 and 4 (this is the same pair written in reverse order)

    Therefore, all 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

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

Claude-3 Sonnet:

    To find all the factor pairs of a number, we need to find all the pairs of numbers that multiply together to give that number.

    For the number 21, we can follow these steps:

    1. Start with 1 and the number itself (21).
    2. Check if 1 and 21 divide evenly into 21. They do, so (1, 21) is a factor pair.
    3. Next, check if 2 and 10.5 (21/2) divide evenly into 21. 2 does, but 10.5 does not (since 21 is not divisible by 10.5).
    4. Next, check if 3 and 7 (21/3) divide evenly into 21. They both do, so (3, 7) is a factor pair.

    Continuing this process, we find that the factor pairs of 21 are:

    (1, 21)
    (3, 7)

    Therefore, the factor pairs of 21 are (1, 21) and (3, 7).

    ### Answer: The factor pairs of 21 are (1, 21) and (3, 7).

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

Correct? Yes

🔗 Problem

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

Claude-3 Sonnet:

    To find all the factor pairs of 72, we need to find all the pairs of numbers that multiply together to give 72.

    Here are the steps:
    1) List out all the factors of 72.
    2) Pair up each factor with its corresponding factor pair.

    The factors of 72 are: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72

    Now, we pair up each factor with its corresponding factor pair:
    1 and 72
    2 and 36
    3 and 24
    4 and 18
    6 and 12
    8 and 9

    Therefore, the factor pairs of 72 are:
    1 and 72
    2 and 36
    3 and 24
    4 and 18
    6 and 12
    8 and 9

    ### Answer: The factor pairs of 72 are: 1 and 72, 2 and 36, 3 and 24, 4 and 18, 6 and 12, 8 and 9.

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

Correct? Yes

🔗 Problem

Problem: Determine all the factor pairs of 38.

Claude-3 Sonnet:

    To find all the factor pairs of a number, we need to find all the pairs of numbers that multiply together to give that number.

    Here are the steps to find the factor pairs of 38:

    1. List out all the factors of 38:
       Factors of 38 = 1, 2, 19, 38

    2. Pair up the factors such that their product is 38:
       1 × 38
       2 × 19

    Therefore, the factor pairs of 38 are:
    (1, 38) and (2, 19)

    Note: The order of the factors in each pair does not matter, so (1, 38) and (38, 1) represent the same factor pair.

    ### Answer: The factor pairs of 38 are (1, 38) and (2, 19).

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

Correct? Yes

🔗 Problem

Problem: List all the factor pairs of 94.

Claude-3 Sonnet:

    To find the factor pairs of a number, we need to find all the pairs of numbers that multiply together to give that number.

    Here are the steps to find the factor pairs of 94:

    1. Start with 1 and the number itself (94).
    2. Find the numbers that divide into 94 evenly.
    3. Pair up each divisor with its corresponding quotient.

    The factor pairs of 94 are:

    1 and 94
    2 and 47

    Since 94 is an even number, we can also list the factor pairs in reverse order:

    47 and 2
    94 and 1

    Therefore, the complete list of factor pairs of 94 is:
    1 and 94
    2 and 47
    47 and 2
    94 and 1

    ### Answer: 1 and 94, 2 and 47, 47 and 2, 94 and 1

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

Correct? Yes

🔗 Problem

Problem: List all factor pairs of 32.

Claude-3 Sonnet:

    To find the factor pairs of a number, we need to find all the pairs of numbers that multiply together to give that number.

    The factor pairs of 32 are:

    1 and 32
    2 and 16
    4 and 8

    These are all the factor pairs because:
    1 × 32 = 32
    2 × 16 = 32
    4 × 8 = 32

    Therefore, the list of all factor pairs of 32 is:
    (1, 32)
    (2, 16)
    (4, 8)

    ### Answer: (1, 32), (2, 16), (4, 8)

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

Correct? Yes

🔗 Problem

Problem: Identify all factor pairs of the number 33.

Claude-3 Sonnet:

    To find the factor pairs of a number, we need to find all the pairs of numbers that multiply together to give that number.

    The factors of 33 are:
    1 and 33
    3 and 11

    Therefore, the factor pairs of 33 are:
    (1, 33)
    (3, 11)

    Note that we list the pairs in both orders, (a, b) and (b, a), since multiplication is commutative.

    ### Answer: The factor pairs of 33 are (1, 33) and (3, 11).

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

Correct? Yes

🔗 Problem

Problem: Write down all the factor pairs of 59.

Claude-3 Sonnet:

    To find the factor pairs of 59, we need to find all the pairs of numbers that multiply together to give 59.

    The factors of 59 are:
    1, 59
    2, 29.5 (not a factor pair since 29.5 is not a whole number)
    3, 19.67 (not a factor pair since 19.67 is not a whole number)
    4, 14.75 (not a factor pair since 14.75 is not a whole number)
    5, 11.8 (not a factor pair since 11.8 is not a whole number)
    6, 9.83 (not a factor pair since 9.83 is not a whole number)
    7, 8.43 (not a factor pair since 8.43 is not a whole number)

    Therefore, the factor pairs of 59 are:
    1, 59

    ### Answer: The factor pair of 59 is (1, 59).

Expected answer: [(1, 59)]

Correct? Yes

🔗 Problem

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

Claude-3 Sonnet:

    To find the factor pairs of a number, we need to find all the pairs of numbers that multiply together to give that number.

    The factors of 54 are:
    1, 54
    2, 27
    3, 18
    6, 9

    To list the factor pairs, we pair up each factor with its corresponding factor:

    Factor pairs of 54:
    1 and 54
    2 and 27
    3 and 18
    6 and 9

    ### Answer: The factor pairs of 54 are: (1, 54), (2, 27), (3, 18), (6, 9).

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

Correct? Yes

🔗 Problem

Problem: Determine all factor pairs of the number 85.

Claude-3 Sonnet:

    To find all factor pairs of a number, we need to find all the pairs of numbers that multiply together to give that number.

    Here's how we can find the factor pairs of 85:

    1) First, we list out all the factors of 85:
       Factors of 85 = 1, 5, 17, 85

    2) Next, we pair up the factors such that their product is 85:
       Factor pairs of 85 = (1, 85), (5, 17)

    Note that we can pair up the factors in either order, so (1, 85) and (85, 1) represent the same factor pair.

    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

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

Claude-3 Sonnet:

    To find the factor pairs of a number, we need to find all the pairs of numbers that multiply together to give that number.

    Here are the steps to find the factor pairs of 56:

    1. List out all the factors of 56:
       Factors of 56 = 1, 2, 4, 7, 8, 14, 28, 56

    2. Pair up the factors such that their product is 56:
       1 × 56
       2 × 28
       4 × 14
       7 × 8

    Therefore, the factor pairs of 56 are:
    (1, 56)
    (2, 28)
    (4, 14)
    (7, 8)

    Note: The pairs are listed in the order (smaller factor, larger factor).

    ### 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

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

Claude-3 Sonnet:

    To find the factor pairs of a number, we need to find all the pairs of numbers that multiply together to give that number.

    The factor pairs of 24 are:

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

    These pairs, when multiplied, give the product 24.

    Therefore, the list of all factor pairs of 24 is:
    (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

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

Claude-3 Sonnet:

    To find the factor pairs of a number, we need to find all the pairs of numbers that multiply together to give that number.

    Here are the steps to find the factor pairs of 84:

    1. List out all the factors of 84:
       Factors of 84 = 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84

    2. Pair up the factors such that each pair multiplies to give 84:
       Factor pairs of 84 = (1, 84), (2, 42), (3, 28), (4, 21), (6, 14), (7, 12)

    Therefore, all the factor pairs of the number 84 are:
    (1, 84), (2, 42), (3, 28), (4, 21), (6, 14), (7, 12)

    ### 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

Problem: Identify all factor pairs of the number 36.

Claude-3 Sonnet:

    To find the factor pairs of a number, we need to find all the pairs of numbers that multiply together to give that number.

    The factors of 36 are:
    1, 36
    2, 18
    3, 12
    4, 9
    6, 6

    Therefore, the factor pairs of 36 are:
    (1, 36)
    (2, 18)
    (3, 12)
    (4, 9)
    (6, 6)

    Note that (6, 6) is considered a factor pair because 6 × 6 = 36.

    ### 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

Problem: What are all the factor pairs of 85?

Claude-3 Sonnet:

    To find the factor pairs of a number, we need to find all the pairs of numbers that multiply together to give that number.

    Here are the steps to find the factor pairs of 85:

    1. List out all the factors of 85:
       Factors of 85 = 1, 5, 17, 85

    2. Pair up the factors by dividing 85 by each factor and finding the corresponding pair:
       85 ÷ 1 = 85, so the pair is (1, 85)
       85 ÷ 5 = 17, so the pair is (5, 17)
       85 ÷ 17 = 5, so the pair is (17, 5)
       85 ÷ 85 = 1, so the pair is (85, 1)

    3. The factor pairs of 85 are:
       (1, 85)
       (5, 17)
       (17, 5)
       (85, 1)

    Note that (5, 17) and (17, 5) are the same pair, just written in reverse order.

    ### Answer: The factor pairs of 85 are (1, 85), (5, 17), and (17, 5).

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

Correct? Yes

🔗 Problem

Problem: Identify all factor pairs of the number 34.

Claude-3 Sonnet:

    To find the factor pairs of a number, we need to find all the pairs of numbers that multiply together to give that number.

    The factors of 34 are:
    1, 34
    2, 17
    Reasoning:
    1 x 34 = 34
    2 x 17 = 34

    Therefore, the factor pairs of 34 are:
    (1, 34)
    (2, 17)

    Note that we can also consider the reverse pairs (34, 1) and (17, 2), but these are just the same pairs written in reverse order.

    ### Answer: The factor pairs of 34 are (1, 34) and (2, 17).

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

Correct? Yes

🔗 Problem

Problem: Determine all factor pairs of 88.

Claude-3 Sonnet:

    To find all factor pairs of a number, we need to find all the pairs of numbers that multiply together to give that number.

    Here are the steps to find the factor pairs of 88:

    1. List out all the factors of 88.
       The factors of 88 are: 1, 2, 4, 8, 11, 22, 44, 88

    2. Pair up each factor with its corresponding factor that multiplies to 88.
       1 × 88
       2 × 44
       4 × 22
       8 × 11

    Therefore, the factor pairs of 88 are:
    (1, 88)
    (2, 44)
    (4, 22)
    (8, 11)

    Note: The pairs are listed in the order (smaller factor, larger factor).

    ### Answer: The factor pairs of 88 are (1, 88), (2, 44), (4, 22), (8, 11).

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

Correct? Yes

🔗 Problem

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

Claude-3 Sonnet:

    To find all the factor pairs of a number, we need to find all the pairs of numbers that multiply together to give that number.

    Here are the steps to find the factor pairs of 83:

    1. Start with 1 and 83 as the first pair of factors.
    2. Check if any number between 1 and the square root of 83 (approximately 9.11) divides 83 evenly.
    3. If a number divides 83 evenly, pair it with the quotient obtained by dividing 83 by that number.

    Following these steps, we get the factor pairs of 83 as:

    1 and 83
    83 and 1

    Now, checking for numbers between 1 and 9 that divide 83 evenly:
    2 does not divide 83 evenly.
    3 does not divide 83 evenly.
    4 does not divide 83 evenly.
    5 does not divide 83 evenly.
    6 does not divide 83 evenly.
    7 does not divide 83 evenly.
    8 does not divide 83 evenly.
    9 does not divide 83 evenly.

    Therefore, the only factor pairs of 83 are 1 and 83.

    ### Answer: The factor pairs of 83 are: (1, 83)

Expected answer: [(1, 83)]

Correct? Yes

🔗 Problem

Problem: What are all the factor pairs of 60?

Claude-3 Sonnet:

    To find the factor pairs of a number, we need to find all the pairs of numbers that multiply together to give that number.

    Here are the steps to find the factor pairs of 60:

    1. List out all the factors of 60:
       Factors of 60 = 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60

    2. Pair up the factors such that their product is 60:
       1 × 60
       2 × 30
       3 × 20
       4 × 15
       5 × 12
       6 × 10

    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

Problem: Write down all factor pairs of 58.

Claude-3 Sonnet:

    To find the factor pairs of 58, we need to find all the pairs of numbers that multiply together to give 58.

    The factors of 58 are:
    1 and 58
    2 and 29

    Therefore, the factor pairs of 58 are:
    1, 58
    2, 29
    29, 2
    58, 1

    ### Answer: The factor pairs of 58 are: (1, 58), (2, 29), (29, 2), (58, 1).

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

Correct? Yes

🔗 Problem

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

Claude-3 Sonnet:

    To find the factor pairs of a number, we need to find all the pairs of numbers that multiply together to give that number.

    The factor pairs of 40 are:

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

    These are all the factor pairs because:
    1 × 40 = 40
    2 × 20 = 40
    4 × 10 = 40
    5 × 8 = 40

    ### Answer: The factor pairs of 40 are: (1, 40), (2, 20), (4, 10), (5, 8).

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

Correct? Yes

🔗 Problem

Problem: Identify all the factor pairs of 91.

Claude-3 Sonnet:

    To find the factor pairs of 91, we need to find all the pairs of numbers that multiply together to give 91.

    Here's the step-by-step process:

    1. Start with 1 and 91 as the first pair of factors.
    2. Check if 91 is divisible by 2. Since it's not, move to the next number.
    3. Check if 91 is divisible by 3. Since it's not, move to the next number.
    4. Check if 91 is divisible by 4. Since it's not, move to the next number.
    5. Check if 91 is divisible by 5. Since it's not, move to the next number.
    6. Check if 91 is divisible by 6. Since it's not, move to the next number.
    7. Check if 91 is divisible by 7. Yes, 91 ÷ 7 = 13. So, we have the factor pair (7, 13).
    8. Continue this process until we reach the square root of 91, which is approximately 9.54.

    The factor pairs of 91 are:
    (1, 91)
    (7, 13)

    Therefore, all 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

Problem: List all the factor pairs of 76.

Claude-3 Sonnet:

    To find the factor pairs of 76, we need to find all the pairs of numbers that multiply together to give 76.

    The factors of 76 are:
    1, 76
    2, 38
    4, 19

    Therefore, the factor pairs of 76 are:
    1 and 76
    2 and 38
    4 and 19

    Note that each pair is listed in both orders, (1, 76) and (76, 1), (2, 38) and (38, 2), (4, 19) and (19, 4).

    ### Answer: (1, 76), (2, 38), (4, 19)

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

Correct? Yes

🔗 Problem

Problem: What are all the factor pairs of 13?

Claude-3 Sonnet:

    To find the factor pairs of a number, we need to find all the pairs of numbers that multiply together to give that number.

    The factors of 13 are:
    1 and 13

    Since 13 is a prime number, it has no other factors besides 1 and itself.

    Therefore, the only factor pair of 13 is:
    1 and 13

    We can represent this factor pair as (1, 13).

    ### Answer: (1, 13)

Expected answer: [(1, 13)]

Correct? Yes

🔗 Problem

Problem: Find all factor pairs of 64.

Claude-3 Sonnet:

    To find all factor pairs of a number, we need to find all the pairs of numbers that multiply together to give that number.

    For 64, we can start by finding the factors:
    1 x 64 = 64
    2 x 32 = 64
    4 x 16 = 64
    8 x 8 = 64

    These are all the factor pairs of 64.

    We can list them out as follows:
    (1, 64)
    (2, 32)
    (4, 16)
    (8, 8)

    Note that (8, 8) is the same as (8, 8), so we only need to list it once.

    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

Problem: What are all factor pairs of 8?

Claude-3 Sonnet:

    To find the factor pairs of a number, we need to find all the pairs of numbers that multiply together to give that number.

    The factors of 8 are: 1, 2, 4, 8

    The factor pairs of 8 are:
    1 and 8
    2 and 4

    Therefore, all the factor pairs of 8 are:
    (1, 8)
    (2, 4)

    ### Answer: (1, 8), (2, 4)

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

Correct? Yes

🔗 Problem

Problem: List all the factor pairs of 80.

Claude-3 Sonnet:

    To find the factor pairs of a number, we need to find all the pairs of numbers that multiply together to give that number.

    The factors of 80 are:
    1, 80
    2, 40
    4, 20
    5, 16
    8, 10

    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

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

Claude-3 Sonnet:

    To find all the factor pairs for a number, we need to find all the pairs of numbers that multiply together to give that number.

    Here are the steps to find the factor pairs of 96:

    1. Start with 1 and the number itself (96).
       Factor pair: 1 and 96

    2. Check if 96 is divisible by 2.
       96 ÷ 2 = 48
       Factor pair: 2 and 48

    3. Check if 96 is divisible by 3.
       96 ÷ 3 = 32
       Factor pair: 3 and 32

    4. Check if 96 is divisible by 4.
       96 ÷ 4 = 24
       Factor pair: 4 and 24

    5. Check if 96 is divisible by 6.
       96 ÷ 6 = 16
       Factor pair: 6 and 16

    6. Check if 96 is divisible by 8.
       96 ÷ 8 = 12
       Factor pair: 8 and 12

    7. Stop when the factors become repetitive.

    Therefore, the factor pairs of 96 are:
    1 and 96
    2 and 48
    3 and 32
    4 and 24
    6 and 16
    8 and 12

    ### Answer: The factor pairs of 96 are: 1 and 96, 2 and 48, 3 and 32, 4 and 24, 6 and 16, 8 and 12.

Expected answer: [(1, 96), (2, 48), (3, 32), (4, 24), (6, 16), (8, 12)]

Correct? Yes

🔗 Problem

Problem: Calculate all factor pairs of 19.

Claude-3 Sonnet:

    To find all factor pairs of a number, we need to find all the pairs of numbers that multiply together to give that number.

    For 19, the factor pairs are:

    1 and 19
    19 and 1

    These are the only factor pairs of 19 because 19 is a prime number, and prime numbers have only two factors: 1 and the number itself.

    Therefore, the factor pairs of 19 are:
    (1, 19)
    (19, 1)

    ### Answer: (1, 19), (19, 1)

Expected answer: [(1, 19)]

Correct? Yes

🔗 Problem

Problem: What are all factor pairs of 37?

Claude-3 Sonnet:

    To find all the factor pairs of a number, we need to find all the pairs of numbers that multiply together to give that number.

    The factors of 37 are:
    1 and 37 (1 × 37 = 37)

    Since 37 is a prime number, it has no other factors besides 1 and itself.

    Therefore, the only factor pair of 37 is (1, 37).

    ### Answer: (1, 37)

Expected answer: [(1, 37)]

Correct? Yes

🔗 Problem

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

Claude-3 Sonnet:

    To find the factor pairs of a number, we need to find all the pairs of numbers that multiply together to give that number.

    The factor pairs of 6 are:

    1 and 6
    2 and 3

    These are the only two factor pairs of 6 because no other pairs of numbers multiply together to give 6.

    Therefore, 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

Problem: List down all the factor pairs of 58.

Claude-3 Sonnet:

    To find the factor pairs of 58, we need to find all the pairs of numbers that multiply together to give 58.

    Here are the steps:
    1) Start with 1 and 58 as the first pair.
    2) Find the next pair by dividing 58 by the next number greater than 1.
    3) Continue this process until the two numbers in the pair are equal.

    The factor pairs of 58 are:
    1, 58
    2, 29

    Since 58 is an even number, we can stop here because the remaining factors will be the same pairs in reverse order (29, 2 and 58, 1).

    Therefore, the factor pairs of 58 are:
    1, 58
    2, 29
    29, 2

    ### Answer: The factor pairs of 58 are: 1, 58; 2, 29; 29, 2.

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

Correct? Yes

🔗 Problem

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

Claude-3 Sonnet:

    To find the factor pairs of a number, we need to find all the pairs of numbers that multiply together to give that number.

    The factors of 81 are:
    1 and 81
    3 and 27
    9 and 9

    Therefore, the factor pairs of 81 are:
    (1, 81)
    (3, 27)
    (9, 9)

    Note that (9, 9) is considered a factor pair because a number is a factor of itself.

    ### Answer: (1, 81), (3, 27), (9, 9)

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

Correct? Yes

🔗 Problem

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

Claude-3 Sonnet:

    To find the factor pairs of a number, we need to find all the pairs of numbers that multiply together to give that number.

    Here are the steps to find the factor pairs of 85:

    1. Start with 1 and the number itself (85).
    2. Find the numbers that divide evenly into 85.
    3. Pair up each divisor with its corresponding quotient.

    The factor pairs of 85 are:

    1 and 85
    5 and 17

    Therefore, the list of all factor pairs of 85 is:
    (1, 85)
    (5, 17)
    (17, 5)
    (85, 1)

    ### Answer: (1, 85), (5, 17), (17, 5), (85, 1)

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

Correct? Yes

🔗 Problem

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

Claude-3 Sonnet:

    To find all the factor pairs of 93, we need to find all the pairs of numbers that multiply together to give 93.

    The factors of 93 are:
    1, 93
    3, 31

    Therefore, the factor pairs of 93 are:
    (1, 93)
    (3, 31)

    Note that we list the pairs in both orders, (a, b) and (b, a), since multiplication is commutative.

    ### Answer: (1, 93), (3, 31)

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

Correct? Yes

🔗 Problem

Problem: Identify all factor pairs of 24.

Claude-3 Sonnet:

    To find the factor pairs of 24, we need to find all the pairs of numbers that multiply together to give 24.

    The factors of 24 are: 1, 2, 3, 4, 6, 8, 12, and 24.

    The factor pairs of 24 are:

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

    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

Problem: List all the factor pairs of 97.

Claude-3 Sonnet:

    To find the factor pairs of a number, we need to find all the pairs of numbers that multiply together to give that number.

    Here are the steps to find the factor pairs of 97:

    1. Start with 1 and 97 as the first pair of factors.
    2. Check if any number between 1 and √97 (approximately 9.85) divides 97 evenly.
    3. If a number divides 97 evenly, pair it with the quotient obtained by dividing 97 by that number.

    Following these steps, we get the factor pairs of 97 as:

    1 and 97
    97 and 1

    There are no other factor pairs because 97 is a prime number, and prime numbers have only two factors: 1 and the number itself.

    Therefore, the factor pairs of 97 are:
    1, 97
    97, 1

    ### Answer: The factor pairs of 97 are 1, 97 and 97, 1.

Expected answer: [(1, 97)]

Correct? Yes

🔗 Problem

Problem: Identify all factor pairs of the number 57.

Claude-3 Sonnet:

    To find all factor pairs of a number, we need to find all the pairs of numbers that multiply together to give that number.

    Here are the steps to find the factor pairs of 57:

    1. List out all the factors of 57.
       The factors of 57 are: 1, 3, 19, 57

    2. Pair up each factor with its corresponding factor pair.
       1 × 57
       3 × 19

    Therefore, the factor pairs of 57 are:
    (1, 57)
    (3, 19)

    Note: The pairs are listed in both orders, (a, b) and (b, a), since multiplication is commutative.

    ### Answer: The factor pairs of 57 are (1, 57) and (3, 19).

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

Correct? Yes

🔗 Problem

Problem: List down all the factor pairs of 19.

Claude-3 Sonnet:

    To find the factor pairs of a number, we need to find all the pairs of numbers that multiply together to give that number.

    The factors of 19 are:
    1 and 19

    To find the factor pairs, we need to pair up the factors:
    (1, 19)

    Therefore, the factor pair of 19 is:
    (1, 19)

    ### Answer: (1, 19)

Expected answer: [(1, 19)]

Correct? Yes

🔗 Problem

Problem: What are all factor pairs of 38?

Claude-3 Sonnet:

    To find all the factor pairs of a number, we need to find all the pairs of numbers that multiply together to give that number.

    Here are the steps to find the factor pairs of 38:

    1. List out all the factors of 38:
       Factors of 38 = 1, 2, 19, 38

    2. Pair up the factors such that their product is 38:
       Factor pairs of 38 = (1, 38), (2, 19)

    Therefore, the factor pairs of 38 are:
    (1, 38) and (2, 19)

    ### Answer: (1, 38), (2, 19)

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

Correct? Yes

🔗 Problem

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

Claude-3 Sonnet:

    To find the factor pairs of a number, we need to find all the pairs of numbers that multiply together to give that number.

    For the number 6, the factor pairs are:

    1 and 6
    2 and 3

    These are the only two pairs of numbers that multiply together to give 6.

    Explanation:
    1 × 6 = 6
    2 × 3 = 6

    Therefore, the factor pairs of 6 are (1, 6) and (2, 3).

    ### Answer: The factor pairs of 6 are (1, 6) and (2, 3).

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

Correct? Yes

🔗 Problem

Problem: Write down all factor pairs of 17.

Claude-3 Sonnet:

    To find the factor pairs of 17, we need to find all the numbers that divide evenly into 17.

    The factors of 17 are:
    1 and 17

    Since 17 is a prime number, it has no other factors besides 1 and itself.

    Therefore, the only factor pair of 17 is:
    1, 17

    ### Answer: (1, 17)

Expected answer: [(1, 17)]

Correct? Yes

🔗 Problem

Problem: List all factor pairs of 45.

Claude-3 Sonnet:

    To find the factor pairs of 45, we need to find all the pairs of numbers that multiply together to give 45.

    The factors of 45 are:
    1, 45
    3, 15
    5, 9

    Therefore, the factor pairs of 45 are:
    (1, 45)
    (3, 15)
    (5, 9)
    (45, 1)
    (15, 3)
    (9, 5)

    ### Answer: (1, 45), (3, 15), (5, 9), (45, 1), (15, 3), (9, 5)

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

Correct? Yes

🔗 Problem

Problem: Write down all factor pairs of 36.

Claude-3 Sonnet:

    To find the factor pairs of 36, we need to find all the pairs of numbers that multiply together to give 36.

    The factor pairs of 36 are:

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

    Note that 6 and 6 is the same pair, so we only list it once.

    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

Problem: What are all factor pairs of 89?

Claude-3 Sonnet:

    To find all factor pairs of 89, we need to find all the numbers that divide into 89 evenly, along with their corresponding pairs.

    The factors of 89 are:
    1 and 89
    89 and 1

    Now, we need to find the remaining factor pairs by dividing 89 by each number from 2 to the square root of 89 (which is approximately 9.43) and checking if the division is exact (i.e., the remainder is zero).

    2 goes into 89, 44 times with a remainder of 1, so 2 is not a factor.
    3 goes into 89, 29 times with a remainder of 2, so 3 is not a factor.
    4 goes into 89, 22 times with a remainder of 1, so 4 is not a factor.
    5 goes into 89, 17 times with a remainder of 4, so 5 is not a factor.
    6 goes into 89, 14 times with a remainder of 5, so 6 is not a factor.
    7 goes into 89, 12 times with a remainder of 5, so 7 is not a factor.
    8 goes into 89, 11 times with a remainder of 1, so 8 is not a factor.
    9 goes into 89, 9 times with a remainder of 8, so 9 is not a factor.

    Therefore, the only factor pairs of 89 are:
    1 and 89
    89 and 1

    ### Answer: The factor pairs of 89 are 1 and 89, 89 and 1.

Expected answer: [(1, 89)]

Correct? Yes

🔗 Problem

Problem: List all the factor pairs of 89.

Claude-3 Sonnet:

    To find the factor pairs of 89, we need to find all the pairs of numbers that multiply together to give 89.

    The factors of 89 are:
    1, 89
    89, 1
    7, 89/7 = 12.71... (not a whole number, so 7 and 89/7 are not a factor pair)
    89/7 = 12.71..., 7 (not a whole number, so 89/7 and 7 are not a factor pair)

    Therefore, the only factor pair of 89 is 1 and 89.

    ### Answer: (1, 89)

Expected answer: [(1, 89)]

Correct? Yes

🔗 Problem

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

Claude-3 Sonnet:

    To find the factor pairs of a number, we need to find all the pairs of numbers that multiply together to give that number.

    The factors of 44 are:
    1, 44
    2, 22
    4, 11

    Therefore, the factor pairs of 44 are:
    1 and 44
    2 and 22
    4 and 11

    ### 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

Problem: Identify all factor pairs of the number 36.

Claude-3 Sonnet:

    To find the factor pairs of a number, we need to find all the pairs of numbers that multiply together to give that number.

    The factors of 36 are:
    1, 36
    2, 18
    3, 12
    4, 9
    6, 6

    Therefore, the factor pairs of 36 are:
    (1, 36)
    (2, 18)
    (3, 12)
    (4, 9)
    (6, 6)

    Note that (6, 6) is considered a factor pair because 6 × 6 = 36.

    ### 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

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

Claude-3 Sonnet:

    To find the factor pairs of a number, we need to find all the pairs of numbers that multiply together to give that number.

    The factors of 35 are: 1, 5, 7, and 35 (because 1 × 35 = 35, 5 × 7 = 35).

    The factor pairs of 35 are:
    1 and 35
    5 and 7

    Therefore, the list of all possible factor pairs of 35 is:
    (1, 35)
    (5, 7)

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

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

Correct? Yes

🔗 Problem

Problem: Write down all factor pairs of 22.

Claude-3 Sonnet:

    To find the factor pairs of 22, we need to find all the pairs of numbers that multiply together to give 22.

    The factors of 22 are:
    1, 22
    2, 11

    Therefore, the factor pairs of 22 are:
    (1, 22)
    (2, 11)

    ### Answer: (1, 22), (2, 11)

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

Correct? Yes

🔗 Problem

Problem: Find all the factor pairs of 40.

Claude-3 Sonnet:

    To find all the factor pairs of a number, we need to find all the pairs of numbers that multiply together to give that number.

    Here are the steps to find the factor pairs of 40:

    1. List out all the factors of 40.
       The factors of 40 are: 1, 2, 4, 5, 8, 10, 20, 40

    2. Pair up each factor with its corresponding factor that multiplies to 40.
       1 × 40
       2 × 20
       4 × 10
       5 × 8

    Therefore, the factor pairs of 40 are:
    1 and 40
    2 and 20
    4 and 10
    5 and 8

    ### Answer: The factor pairs of 40 are: (1, 40), (2, 20), (4, 10), (5, 8).

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

Correct? Yes

🔗 Problem

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

Claude-3 Sonnet:

    To find the factor pairs of a number, we need to find all the pairs of numbers that multiply together to give that number.

    The factor pairs of 64 are:

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

    So, the list of all factor pairs of 64 is:
    (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

Problem: Determine all factor pairs of 30.

Claude-3 Sonnet:

    To find all factor pairs of a number, we need to find all the pairs of numbers that multiply together to give that number.

    For 30, we can follow these steps:

    1. Start with 1 and 30 as the first pair of factors.
    2. Divide 30 by 2 to get 15. Now we have two pairs of factors: (1, 30) and (2, 15).
    3. Divide 15 by 3 to get 5. Now we have three pairs of factors: (1, 30), (2, 15), and (3, 10).
    4. Divide 10 by 2 to get 5. Now we have four pairs of factors: (1, 30), (2, 15), (3, 10), and (5, 6).

    Since 6 is the square root of 30, we have found all the factor pairs.

    Therefore, 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

Problem: List all factor pairs of 59.

Claude-3 Sonnet:

    To find the factor pairs of 59, we need to find all the numbers that divide into 59 evenly, along with their corresponding pairs.

    The factors of 59 are:
    1 and 59
    59 and 1

    To find the remaining factor pairs, we can check the numbers from 2 to the square root of 59 (approximately 7.68).

    The factor pairs of 59 are:
    1 and 59
    59 and 1

    Therefore, the list of all factor pairs of 59 is:
    1, 59
    59, 1

    ### Answer: (1, 59), (59, 1)

Expected answer: [(1, 59)]

Correct? Yes

🔗 Problem

Problem: List all factor pairs of 18.

Claude-3 Sonnet:

    To find the factor pairs of a number, we need to find all the pairs of numbers that multiply together to give that number.

    The factors of 18 are: 1, 2, 3, 6, 9, 18.

    The factor pairs of 18 are:
    1 and 18
    2 and 9
    3 and 6

    Therefore, the list of all factor pairs of 18 is:
    (1, 18)
    (2, 9)
    (3, 6)

    ### Answer: (1, 18), (2, 9), (3, 6)

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

Correct? Yes

🔗 Problem

Problem: Please list all the factor pairs of 61.

Claude-3 Sonnet:

    To find the factor pairs of 61, we need to find all the pairs of numbers that multiply together to give 61.

    The factors of 61 are:
    1 and 61
    61 and 1

    To find the remaining factor pairs, we can divide 61 by each number from 2 to the square root of 61 (which is approximately 7.81) and check if the division is exact (i.e., the remainder is zero). If so, we have found a factor pair.

    Here are the factor pairs of 61:
    1 and 61
    61 and 1

    Therefore, the list of all factor pairs of 61 is:
    1, 61
    61, 1

    ### Answer: (1, 61), (61, 1)

Expected answer: [(1, 61)]

Correct? Yes

🔗 Problem

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

Claude-3 Sonnet:

    To find the factor pairs of a number, we need to find all the pairs of numbers that multiply together to give that number.

    Here are the steps to find the factor pairs of 86:

    1. List out all the factors of 86:
       Factors of 86 = 1, 2, 43, 86

    2. Pair up the factors such that each pair multiplies to give 86:
       Factor pairs of 86 = (1, 86), (2, 43)

    Therefore, the factor pairs of 86 are:
    (1, 86) and (2, 43)

    ### Answer: (1, 86), (2, 43)

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

Correct? Yes

🔗 Problem

Problem: List out all factor pairs of 57.

Claude-3 Sonnet:

    To find the factor pairs of 57, we need to find all the numbers that divide evenly into 57.

    Here are the steps:
    1) Start with 1 and 57 as the first factor pair.
    2) Find other factors by dividing 57 by consecutive numbers starting from 2.
    3) If the division is exact (with no remainder), then the divisor and the quotient form a factor pair.
    4) Stop when the divisor becomes greater than the square root of 57 (approximately 7.55).

    The factor pairs of 57 are:
    1 and 57
    3 and 19
    19 and 3
    57 and 1

    Therefore, the list of all factor pairs of 57 is:
    (1, 57), (3, 19), (19, 3)

    ### Answer: (1, 57), (3, 19), (19, 3)

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

Correct? Yes

🔗 Problem

Problem: What are all factor pairs of 38?

Claude-3 Sonnet:

    To find all the factor pairs of a number, we need to find all the pairs of numbers that multiply together to give that number.

    Here are the steps to find the factor pairs of 38:

    1. List out all the factors of 38:
       Factors of 38 = 1, 2, 19, 38

    2. Pair up the factors such that their product is 38:
       Factor pairs of 38 = (1, 38), (2, 19)

    Therefore, the factor pairs of 38 are:
    (1, 38) and (2, 19)

    ### Answer: (1, 38), (2, 19)

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

Correct? Yes

🔗 Problem

Problem: Please list all factor pairs of 27.

Claude-3 Sonnet:

    To find the factor pairs of 27, we need to find all the pairs of numbers that multiply together to give 27.

    The factors of 27 are: 1, 3, 9, 27.

    The factor pairs of 27 are:

    1 and 27
    3 and 9

    Therefore, the list of all factor pairs of 27 is:
    (1, 27)
    (3, 9)

    ### Answer: (1, 27), (3, 9)

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

Correct? Yes

🔗 Problem

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

Claude-3 Sonnet:

    To find the factor pairs of a number, we need to find all the pairs of numbers that multiply together to give that number.

    Here are the steps to find the factor pairs of 78:

    1. Start with 1 and the number itself (78).
       Factor pair: 1 and 78

    2. Check if 78 is divisible by 2.
       78 ÷ 2 = 39
       Factor pair: 2 and 39

    3. Check if 78 is divisible by 3.
       78 ÷ 3 = 26
       Factor pair: 3 and 26

    4. Check if 78 is divisible by 4.
       It is not divisible by 4.

    5. Check if 78 is divisible by 5.
       It is not divisible by 5.

    6. Check if 78 is divisible by 6.
       78 ÷ 6 = 13
       Factor pair: 6 and 13

    The factor pairs of 78 are:
    1 and 78
    2 and 39
    3 and 26
    6 and 13

    ### Answer: The factor pairs of 78 are: 1 and 78, 2 and 39, 3 and 26, 6 and 13.

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

Correct? Yes

🔗 Problem

Problem: List out all the factor pairs of 23.

Claude-3 Sonnet:

    To find the factor pairs of 23, we need to find all the pairs of numbers that multiply together to give 23.

    The factors of 23 are:
    1, 23

    Now, we need to find the pairs of factors that multiply to give 23:
    1 × 23 = 23

    Therefore, the factor pairs of 23 are:
    (1, 23)

    Note that 23 is a prime number, so it only has one factor pair (1 and itself).

    ### Answer: (1, 23)

Expected answer: [(1, 23)]

Correct? Yes

🔗 Problem

Problem: List down all the factor pairs of 2.

Claude-3 Sonnet:

    The factor pairs of 2 are:

    1 and 2

    Since 2 is a prime number, it only has one factor pair, which is 1 and itself (2).

    ### Answer: (1, 2)

Expected answer: [(1, 2)]

Correct? Yes

🔗 Problem

Problem: Write down all the factor pairs of 49.

Claude-3 Sonnet:

    To find the factor pairs of 49, we need to find all the pairs of numbers that multiply together to give 49.

    The factors of 49 are:
    1, 49
    7, 7

    Therefore, the factor pairs of 49 are:
    (1, 49)
    (7, 7)

    Note that 7 is repeated because it is a factor of itself.

    ### Answer: (1, 49), (7, 7)

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

Correct? Yes

🔗 Problem

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

Claude-3 Sonnet:

    To find the distinct pairs of factors of 93, we need to find all the pairs of numbers that multiply to give 93.

    First, let's list out all the factors of 93:
    1, 3, 31, 93

    Now, we can pair up these factors to form distinct pairs:

    1 × 93
    3 × 31

    Note that we don't need to include the pair 93 × 1 since it's the same as 1 × 93.

    Therefore, the distinct pairs of factors that can multiply to give the number 93 are:
    1 × 93
    3 × 31

    ### Answer: (1, 93), (3, 31)

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

Correct? Yes