# 133 and Level 6

133 is a composite number. 133 = 1 x 133 or 7 x 19. Factors of 133: 1, 7, 19, 133. Prime factorization 133 = 7 x 19.

• The divisibility trick for 7 tells us to split 133 into 13 and 3. We double 3 and find the difference between the double and 13. We get 13 (2 x 3)  = 13 – 6  = 7 (an obvious multiple of 7) so 7 is a factor of 133.

133 is never a clue in the FIND THE FACTORS puzzles.

Excel file of puzzles and previous week’s solutions: 10 Factors 2014-05-26

# 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

# 131 and Level 4

• 131 is a prime number.
• Prime factorization: 131 is prime.
• The exponent of prime number 131 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 131 has exactly 2 factors.
• Factors of 131: 1, 131
• Factor pairs: 131 = 1 x 131
• 131 has no square factors that allow its square root to be simplified. √131 ≈ 11.4455

How do we know that 131 is a prime number? If 131 were not a prime number, then it would be divisible by at least one prime number less than or equal to √131 ≈ 11.4. Since 131 cannot be divided evenly by 2, 3, 5, 7, or 11, we know that 131 is a prime number.

131 is never a clue in the FIND THE FACTORS puzzles.

Excel file of puzzles and previous week’s solutions: 10 Factors 2014-05-26

# 130 and Level 3

130 is a composite number. 130 = 1 x 130, 2 x 65, 5 x 26, or 10 x 13. Factors of 130: 1, 2, 5, 10, 13, 26, 65, 130. Prime factorization: 130 = 2 x 5 x 13.

130 is never a clue in the  FIND THE FACTORS 1 – 10 or 1 – 12 puzzles.

Excel file of puzzles and previous week’s solutions: 10 Factors 2014-05-26

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:

# 129 and Level 2

129  is a composite number. 129 = 1 x 129 or 3 x 43. Factors of 129: 1, 3, 43, 129. Prime factorization: 129 = 3 x 43.

2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 = 129, thus 129 is the sum of the first 10 prime numbers.

129 is never a clue in the FIND THE FACTORS puzzles.

Excel file of puzzles and previous week’s solutions: 10 Factors 2014-05-26

# 128 and Level 1

128  is a composite number. 128 = 1 x 128, 2 x 64, 4 x 32, or 8 x 16. Factors of 128: 1, 2, 4, 8, 16, 32, 64, 128. Prime factorization: 128 = 2×2×2×2×2×2×2 which can be written 128 = 2⁷

128 is never a clue in the FIND THE FACTORS  puzzles.

Excel file of puzzles and last week’s solutions: 10 Factors 2014-05-26

# 127 and Level 6

• 127 is a prime number.
• Prime factorization: 127 is prime.
• The exponent of prime number 127 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 127 has exactly 2 factors.
• Factors of 127: 1, 127
• Factor pairs: 127 = 1 x 127
• 127 has no square factors that allow its square root to be simplified. √127 ≈ 11.2694

How do we know that 127 is a prime number? If 127 were not a prime number, then it would be divisible by at least one prime number less than or equal to √127 ≈ 11.3. Since 127 cannot be divided evenly by 2, 3, 5, 7, or 11, we know that 127 is a prime number.

Thinking process using divisibility tricks to prove that 127 is not divisible by 2, 3, 5, 7, or 11:

• 127 is not even, so 127 is not divisible by 2.
• 1 + 2 + 7 = 10 which is not divisible by 3, so 127 is not divisible by 3.
• The last digit of 127 is not 5 or 0, so 127 is not divisible by 5.
• The divisibility trick for 7 requires us to split 127 into 12 and 7. We double 7 and find the difference between the double and 12. We get (2 x 7) – 12  = 14 – 12 = 2. Since 2 is not divisible by 7, 127 is also not divisible by 7.
• To perform the divisibility trick for 11 we look at the digits another way: Digits that are in an odd position are in regular type and the digit in an even position is in bold type: 127. We find the sum of the numbers typed in regular type (1 + 7  = 8) and the sum of the number(s) typed in bold type (2) and subtract the smaller sum from the larger sum (8 – 2 = 6). If the difference were a multiple of 11, than the original number would also be a  multiple of 11 (and divisible by 11). Since 6 is not a multiple of 11, we know 127 is not divisible by 11.

Therefore the only factor pair that exists for 127 is 1 x 127, and it is a prime number.

127 is never a clue in the FIND THE FACTORS puzzles.

Spotting the logic needed to begin this Level 6 puzzle should be easier than usual. If you have completed Level 5 puzzles, but have never before attempted a Level 6 puzzle, try this one !

Excel file of this week’s puzzles and last week’s factors: 12 Factors 2014-05-19

# 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?

1. Yes, all whole numbers are divisible by 1, so 1 x 126 = 126.
2. 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.)
3. 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.
4. No, the number formed from its last two digits, 26, isn’t divisible by 4, so 126 is not divisible by 4.
5. No, the last digit is not 0 or 5, so 126 is not divisible by 5.
6. Yes, 126 is divisible by both 2 and 3, so it is divisible by 6, and 6 x 21 = 126.
7. 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.
8. No, since 126 isn’t divisible by 4, it is not divisible by any multiple of 4.
9. Yes, see 3 above. 126 is divisible by 9, and 9 x 14 = 126.
10. No, 126 doesn’t ends with a zero, so 10 is not a factor of 126.
11. 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

# 125 and Level 4

125 is a composite number. 125 = 1 x 125 or 5 x 25. Factors of 125: 1, 5, 25, 125. Prime factorization: 125 = 5 x 5 x 5, which can also be written 125 = 5³.

125 is never a clue in the FIND THE FACTORS  puzzles.

Placing the last few factors on this Level 4 puzzle may be a bit trickier than usual. As always, use logic, not guessing and checking, to find its one and only solution. Good luck!

Excel file of this week’s puzzles and last week’s factors: 12 Factors 2014-05-19

# 124 and Level 3

124 is a composite number. 124 = 1 x 124, 2 x 62, or 4 x 31. Factors of 124: 1, 2, 4, 31, 62, 124. Prime factorization:124 = 2 x 2 x 31, which can also be written 124 = 2² x 31.

124 is never a clue in the FIND THE FACTORS puzzles.

Excel file of this week’s puzzles and last week’s factors: 12 Factors 2014-05-19

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: