| 178 is a composite number. |
| Factor pairs: 178 = 1 x 178 or 2 x 89 |
| Factors of 178: 1, 2, 89, 178 |
| Prime factorization: 178 = 2 x 89 |
Excel file of puzzles and previous week’s factor solutions: 12 Factors 2014-07-14
Level 5 Puzzle
172 and Level 5
| 172 is a composite number. |
| Factor pairs: 172 = 1 x 172, 2 x 86, or 4 x 43 |
| Factors of 172: 1, 2, 4, 43, 86, 172 |
| Prime factorization: 172 = 2 x 2 x 43, which can be written (2^2) x 43 |
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Excel file of puzzles and previous week’s factor solutions: 10 Factors 2014-07-07
165 and Level 5
| 165 is a composite number. |
| Factor pairs: 165 = 1 x 165, 3 x 55, 5 x 33, 11 x 15 |
| Factors of 165: 1, 3, 5, 11, 15, 33, 55, 165 |
| Prime factorization: 165 = 3 x 5 x 11 |
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Excel file of puzzles and previous week’s factor solutions: 12 Factors 2014-06-30
159 and Level 5
| 159 is a composite number. |
| Factor pairs: 159 = 1 x 159 or 3 x 53 |
| Factors of 159: 1, 3, 53, 159 |
| Prime factorization: 159 = 3 x 53 |
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Excel file of puzzles and previous week’s factor solutions: 10 Factors 2014-06-23
153 and Level 5
| 153 is a composite number and so much more! |
| Factor pairs: 153 = 1 x 153, 3 x 51, or 9 x 17. |
| Factors of 153: 1, 3, 9, 17, 51, 153. |
| Prime factorization: 153 = 3 x 3 x 17, which can be written (3^2) x 17. |
I first learned that 153 = 1³ + 5³ + 3³ when I read mystery-numbers-370-153. This “St. Peter’s fish” number is so specific that it baffles scholars and students of the Bible.
Is the fact that 153 = 3•51 just as mysterious?
Excel file of puzzles and previous week’s factor solutions: 12 Factors 2014-06-16
146 and Level 5
146 is a composite number. Factor pairs: 146 = 1 x 146 or 2 x 73. Factors of 146: 1, 2, 73, 146. Prime factorization: 146 = 2 x 73.
Excel file of puzzles and previous week’s factor solutions: 10 Factors 2014-06-09
139 and Level 5
- 139 is a prime number.
- Prime factorization: 139 is prime.
- The exponent of prime number 139 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 139 has exactly 2 factors.
- Factors of 139: 1, 139
- Factor pairs: 139 = 1 x 139
- 139 has no square factors that allow its square root to be simplified. √139 ≈ 11.789826
How do we know that 139 is a prime number? If 139 were not a prime number, then it would be divisible by at least one prime number less than or equal to √139 ≈ 11.8. Since 139 cannot be divided evenly by 2, 3, 5, 7, or 11, we know that 139 is a prime number.
139 is never a clue in the FIND THE FACTORS puzzles.
Excel file with puzzles and the previous week’s factor solutions: 12 Factors 2014-06-02
132 and The Hourglass Is Almost Empty
132 is a composite number. 132 = 1 x 132, 2 x 66, 3 x 44, 4 x 33, 6 x 22, or 11 x 12. Factors of 132: 1, 2, 3, 4, 6, 11, 12, 22, 33, 44, 66, 132. Prime factorization: 132 = 2 x 2 x 3 x 11, which can also be written 132 = 2² x 3 x 11.
Taking the factor pair with the largest square number factor, we get √132 = (√4)(√33) = 2√33 ≈ 11.489
When 132 is a clue in the FIND THE FACTORS 1 – 12 puzzles, always use 11 and 12 as the factors.
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This puzzle reminds me of a couple of hourglass scenes from The Wizard of Oz that can be found in the middle of the page here. Yes, I know my hourglass is about the size of an egg timer, but hurry and solve the puzzle! Dorothy’s life could depend on it.
Excel file of puzzles and previous week’s solutions: 10 Factors 2014-05-26
126 and Level 5
126 is a composite number. 126 = 1 x 126, 2 x 63, 3 x 42, 6 x 21, 7 x 18, or 9 x 14. Factors of 126: 1, 2, 3, 6, 7, 9, 14, 18, 21, 42, 63, 126. Prime factorization: 126 = 2 x 3 x 3 x 7, which can also be written 2 x 3² x 7.
Thinking process using divisibility tricks to find the factor pairs of 126 quickly:
√126 is irrational and approximately equal to 11.22. Every factor pair of 126 will have one factor less than 11.22 and one factor greater than 11.22, and we will find both factors in each pair at the same time. The following integers are less than 11.22. Are they factors of 126?
- Yes, all whole numbers are divisible by 1, so 1 x 126 = 126.
- Yes, 126 is an even number. 126 ÷ 2 = 63, so 2 x 63 = 126. (Since 63 is odd, 4 will not be a factor of 126.)
- Yes, 1 + 2 + 6 = 9 which is divisible by both 3 and 9, so 126 is divisible by both 3 and 9, and 3 x 42 = 126.
- No, the number formed from its last two digits, 26, isn’t divisible by 4, so 126 is not divisible by 4.
- No, the last digit is not 0 or 5, so 126 is not divisible by 5.
- Yes, 126 is divisible by both 2 and 3, so it is divisible by 6, and 6 x 21 = 126.
- Yes, 63, 42, and 21 found above give this fact away. But also try the divisibility trick for 7 which requires us to split 126 into 12 and 6. We double 6 and subtract the double from 12. We get 12 – (2 x 6) = 12 – 12 = 0. Since 0 is divisible by 7, 126 is also divisible by 7, and 7 x 18 = 126.
- No, since 126 isn’t divisible by 4, it is not divisible by any multiple of 4.
- Yes, see 3 above. 126 is divisible by 9, and 9 x 14 = 126.
- No, 126 doesn’t ends with a zero, so 10 is not a factor of 126.
- No. I’ve typed the digits that are in an odd position in regular type and the digit in an even position in bold type: 126. To perform the divisibility trick for 11, we find the sum of the numbers typed in regular type (1 + 6 = 7) and the sum of the numbers typed in bold type (2) and subtract the smaller sum from the larger sum (7 – 2 = 5). If the difference were a multiple of 11, than the original number would also be a multiple of 11 (and divisible by 11). Since 5 is not a multiple of 11, we know 126 is not divisible by 11.
From this thinking process we conclude that the factor pairs of 126 are 1 x 126, 2 x 63, 3 x 42, 6 x 21, 7 x 18, and 9 x 14.
126 is never a clue in the FIND THE FACTORS puzzles.
Today’s puzzle might be the most difficult level 5 puzzle I have ever published, so if you have never solved a level 5 or level 6 puzzle before, you probably don’t want to start with this one.
Excel file of this week’s puzzles and last week’s factors: 12 Factors 2014-05-19
120 and Level 5
Today’s Puzzle:
Write the numbers from 1 to 10 in both the top row and the first column so that this puzzle functions like a multiplication table.
Excel file of this week’s puzzles and last week’s factors: 10 Factors 2014-05-12
Thinking process using divisibility rules to find the factor pairs of 120:
√120 is irrational and approximately equal to 10.95. Every factor pair of 120 will have one factor less than 10.95 and one factor greater than 10.95, and we will find both factors in each pair at the same time. The following numbers are less than 10.95. Are they factors of 120?
- Yes, all whole numbers are divisible by 1, so 1 x 120 = 120.
- Yes, 120 is an even number. 120 ÷ 2 = 60, so 2 x 60 = 120. (Since 60 is even, 4 will also be a factor of 120.)
- Yes, 1 + 2 + 0 = 3 which is divisible by 3 (but not by 9), so 120 is divisible by 3. 120 ÷ 3 = 40, so 3 x 40 = 120. Note 120 will not be divisible by 9.
- Yes, the number formed from its last two digits, 20, is divisible by 4, so 120 is divisible by 4, and 4 x 30 = 120. (Note since 30 is even, 8 will also be a factor of 120.)
- Yes, the last digit is 0 or 5, so 120 is divisible by 5, and 5 x 24 = 120.
- Yes, 120 is divisible by both 2 and 3, so it is divisible by 6, and 6 x 20 = 120.
- No. The divisibility trick for 7 requires us to split 120 into 12 and 0. We double 0 and subtract the double from 12. 12 – (2 x 0) = 12 – 0 = 12. Since 12 is not divisible by 7, 120 also is not divisible by 7.
- Yes, see 4 above. 120 = 8 x 15. (This will mean that ANY number whose last 3 digits are 120 will also be divisible by 8.)
- No, see 3 above. 120 is not divisible by 9.
- Yes, 120 ends with a zero, so 10 is a factor of 120, and 10 x 12 = 120.
From this thinking process we conclude that the factor pairs of 120 are 1 x 120, 2 x 60, 3 x 40, 4 x 30, 5 x 24, 6 x 20, 8 x 15, and 10 x 12.
Factors of 120:
120 is a composite number. 120 = 1 x 120, 2 x 60, 3 x 40, 4 x 30, 5 x 24, 6 x 20, 8 x 15, or 10 x 12. Factors of 120: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120. Prime factorization: 120 = 2 x 2 x 2 x 3 x 5, which can also be written 120 = 2³ x 3 x 5.
When 120 is a clue in the FIND THE FACTORS 1 – 12 puzzles, use 10 and 12 as the factors.
Sum-Difference Puzzle:
30 has four factor pairs. One of those factor pairs adds up to 13, and another one subtracts to 13. Can you write those factors in their proper places in the first puzzle below?
120 has eight factor pairs. One of those factor pairs adds up to 26, and another one subtracts to 26. If you can identify those factor pairs, then you can solve the second puzzle!
The second puzzle is really just the first puzzle in disguise. Why would I say that?
More about the Number 120:
120 = 5! because 1·2·3·4·5 = 120
120 is also the smallest positive multiple of 6 that is neither preceded nor followed by a prime number! (119 = 7 ×1 7, and 121 = 11 × 11, so neither one is prime.)
What kind of shape is 120 in?
- 120 is the 15th triangular number because 15(16)/2 = 120,
it’s the 8th tetrahedral number because (8)(9)(10)/6 = 120 (That means - 120 is the sum of the first eight triangular numbers), and
- it is the 8th hexagonal number because (8)(2·8-1) = 120.
120 is the hypotenuse of a Pythagorean triple:
72-96-120, which is 24 times (3-4-5).
A Logical Way to Find the Solution to Today’s Puzzle:
