115 is a composite number. 115 = 1 x 115 or 5 x 23. Factors of 115: 1, 5, 23, 115. Prime factorization: 115 = 5 x 23.
115 is never a clue in the FIND THE FACTORS puzzles.
Excel file of puzzles and the previous week’s solutions: 12 Factors 2014-05-05
logic
113 and Level 5
- 113 is a prime number.
- Prime factorization: 113 is prime.
- The exponent of prime number 113 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 113 has exactly 2 factors.
- Factors of 113: 1, 113
- Factor pairs: 113 = 1 x 113
- 113 has no square factors that allow its square root to be simplified. √113 ≈ 10.6301
How do we know that 113 is a prime number? If 113 were not a prime number, then it would be divisible by at least one prime number less than or equal to √113 ≈ 10.6 Since 113 cannot be divided evenly by 2, 3, 5, or 7, we know that 113 is a prime number.
113 is never a clue in the FIND THE FACTORS puzzles.
Excel file of puzzles and the previous week’s solutions: 12 Factors 2014-05-05
112 and Level 4
Today’s Puzzle:
Some people get the factors of 54 and 56 mixed up. If you get them mixed up on this particular puzzle, you could still think you solved the puzzle correctly even if you haven’t. Here’s a couple of tricks that might prevent the mix-ups:
One of my favorite multiplication rhymes is “Five, Six, Seven, Eight, Fifty-Six is Seven Times Eight,” so 56 = 7 x 8.
Notice that 5 + 4 = 9. The sum of the digits of any multiple of 9 will always be 9 or another multiple of 9, so 54 = 9 x 6. On the other hand, 5 + 6 = 11, so 56 cannot have 9 as one of its factors.
Excel file of six puzzles and the previous week’s solutions: 12 Factors 2014-05-05
Factors of 112:
112 is a composite number. 112 = 1 x 112, 2 x 56, 4 x 28, 7 x 16, or 8 x 14. Factors of 112: 1, 2, 4, 7, 8, 14, 16, 28, 56, 112. Prime factorization: 112 = 2 x 2 x 2 x 2 x 7, which can also be written 112 = 2⁴ x 7.
More about the Number 112:
112 is never a clue in the FIND THE FACTORS puzzles.
24 is divisible by the sum of its digits and by the product of its digits. So too is 112. Is there a four-digit number with this property? A five-digit one? In general, for each count of digits, is there a number double divisible in this way? pic.twitter.com/PkruSSvx9K
— James Tanton (@jamestanton) March 21, 2021
110 and Level 3
110 is a composite number. 110 = 1 x 110, 2 x 55, 5 x 22, or 10 x 11. Factors of 110: 1, 2, 5, 10, 11, 22, 55, 110. Prime factorization: 110 = 2 x 5 x 11.
When 110 is a clue in the FIND THE FACTORS 1 – 12 puzzles, use 10 and 11 as the factors.
Excel file of puzzles and the previous week’s solutions: 12 Factors 2014-05-05
A Logical Approach to FIND THE FACTORS: Find the column or row with two clues and find their common factor. Write the corresponding factors in the factor column (1st column) and factor row (top row). Because this is a level three puzzle, you have now written a factor at the top of the factor column. Continue to work from the top of the factor column to the bottom, finding factors and filling in the factor column and the factor row one cell at a time as you go:
109 and Level 2
- 109 is a prime number.
- Prime factorization: 109 is prime.
- The exponent of prime number 109 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 109 has exactly 2 factors.
- Factors of 109: 1, 109
- Factor pairs: 109 = 1 x 109
- 109 has no square factors that allow its square root to be simplified. √109 ≈ 10.4403
How do we know that 109 is a prime number? If 109 were not a prime number, then it would be divisible by at least one prime number less than or equal to √109 ≈ 10.4 Since 109 cannot be divided evenly by 2, 3, 5, or 7, we know that 109 is a prime number.
109 is never a clue in the FIND THE FACTORS puzzles.
This week’s puzzles and last weeks solutions: 12 Factors 2014-05-05
108 and Level 1
108 is a composite number. 108 = 1 x 108, 2 x 54, 3 x 36, 4 x 27, 6 x 18, or 9 x 12. Factors of 108: 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108. Prime factorization: 108 = 2 x 2 x 3 x 3 x 3, which can also be written 108 = 2² x 3³
When 108 is a clue in the FIND THE FACTORS 1 – 12 puzzles, use 9 and 12 as the factors.
This week’s puzzles and last weeks solutions: 12 Factors 2014-05-05
107 and Level 6
- 107 is a prime number.
- Prime factorization: 107 is prime.
- The exponent of prime number 107 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 107 has exactly 2 factors.
- Factors of 107: 1, 107
- Factor pairs: 107 = 1 x 107
- 107 has no square factors that allow its square root to be simplified. √107 ≈ 10.34408
How do we know that 107 is a prime number? If 107 were not a prime number, then it would be divisible by at least one prime number less than or equal to √107 ≈ 10.3 Since 107 cannot be divided evenly by 2, 3, 5, or 7, we know that 107 is a prime number.
107 is never a clue in the FIND THE FACTORS puzzles.
This week’s puzzles and last week’s solutions: 10 Factors 2014-04-28
106 and Level 5
106 is a composite number. 106 = 1 x 106 or 2 x 53. Factors of 106: 1, 2, 53, 106. Prime factorization: 106 = 2 x 53.
106 is never a clue in the FIND THE FACTORS puzzles.
This week’s puzzles and last week’s solutions: 10 Factors 2014-04-28
105 and Level 4
105 is a composite number. 105 = 1 x 105, 3 x 35, 5 x 21, 7 x 15. Factors of 105: 1, 3, 5, 7, 15, 21, 35, 105. Prime factorization: 105 = 3 x 5 x 7.
105 is never a clue in the FIND THE FACTORS puzzles.
This week’s puzzles and last week’s solutions: 10 Factors 2014-04-28
104 and Level 3
104 is a composite number. 104 = 1 x 104, 2 x 52, 4 x 26, or 8 x 13. Factors of 104: 1, 2, 4, 8, 13, 26, 52, 104. Prime factorization: 104 = 2 x 2 x 2 x 13, which can also be written 2³ x 13.
104 is never a clue in the FIND THE FACTORS puzzles.
The week’s puzzles and the previous week’s solutions: 10 Factors 2014-04-28
A Logical Approach to find the factors: Find the column or row with two clues and find their common factor. Write the corresponding factors in the factor column and factor row. Because this is a level three puzzle, you have now written a factor at the top of the factor column. Continue to work from the top of the factor column to the bottom, finding factors and filling in the table one row at a time as you go:
