## A Multiplication Based Logic Puzzle

### 538 and Level 2

538 is the hypotenuse of the Pythagorean triple 138-520-538. Can you find the greatest common factor of those three numbers?

Print the puzzles or type the solution on this excel file: 12 Factors 2015-06-29

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• 538 is a composite number.
• Prime factorization: 538 = 2 x 269
• The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 x 2 = 4. Therefore 538 has exactly 4 factors.
• Factors of 538: 1, 2, 269, 538
• Factor pairs: 538 = 1 x 538 or 2 x 269
• 538 has no square factors that allow its square root to be simplified. √538 ≈ 23.194827

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### 537 and Level 1

537 is made from 3 consecutive odd numbers so it is divisible by 3.

Print the puzzles or type the solution on this excel file: 12 Factors 2015-06-29

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• 537 is a composite number.
• Prime factorization: 537 = 3 x 179
• The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 x 2 = 4. Therefore 537 has exactly 4 factors.
• Factors of 537: 1, 3, 179, 537
• Factor pairs: 537 = 1 x 537 or 3 x 179
• 537 has no square factors that allow its square root to be simplified. √537 ≈ 23.17326

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### 536 Family Reunion

Last week I attended a family reunion. My uncle Bob showed me a very clever way that helps him remember the number of children that my dad and each of his siblings had.

In case you are wondering, I was one of Leonard’s fifteen kids: He and his first wife had 4 children. They divorced. He met my mom who already had a child of her own. They married and had 6 children. She died. Then after he married my step-mother who already had two grown children, they had two more.

• 536 is a composite number.
• Prime factorization: 536 = 2 x 2 x 2 x 67, which can be written 536 = (2^3) x 67
• The exponents in the prime factorization are 3 and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1) = 4 x 2 = 8. Therefore 536 has exactly 8 factors.
• Factors of 536: 1, 2, 4, 8, 67, 134, 268, 536
• Factor pairs: 536 = 1 x 536, 2 x 268, 4 x 134, or 8 x 67
• Taking the factor pair with the largest square number factor, we get √536 = (√4)(√134) = 2√134 ≈ 23.15167

### 535 and Level 6

535 is the hypotenuse of the Pythagorean triple 321-428-535. Can you find the greatest common factor of those three numbers?

Print the puzzles or type the solution on this excel file: 10 Factors 2015-06-22

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• 535 is a composite number.
• Prime factorization: 535 = 5 x 107
• The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 x 2 = 4. Therefore 535 has exactly 4 factors.
• Factors of 535: 1, 5, 107, 535
• Factor pairs: 535 = 1 x 535 or 5 x 107
• 535 has no square factors that allow its square root to be simplified. √535 ≈ 23.130067

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### 534 and Level 5

534 is made of three consecutive digits so it can be evenly divided by three.

534 is the sum of consecutive primes: 127 + 131 + 137 + 139 = 534.

534 is the hypotenuse of the Pythagorean triple 234-480-534. Can you find the greatest common factor of those three numbers?

Print the puzzles or type the solution on this excel file: 10 Factors 2015-06-22

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• 534 is a composite number.
• Prime factorization: 534 = 2 x 3 x 89
• The exponents in the prime factorization are 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 x 2 x 2 = 8. Therefore 534 has exactly 8 factors.
• Factors of 534: 1, 2, 3, 6, 89, 178, 267, 534
• Factor pairs: 534 = 1 x 534, 2 x 267, 3 x 178, or 6 x 89
• 534 has no square factors that allow its square root to be simplified. √534 ≈ 23.108440.

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### 533 and Level 4

533 is the sum of consecutive primes two different ways: 173 + 179 + 181 = 533 = 101 + 103 + 107 + 109 + 113.

533 = (23^2) + (2^2), and 533 = (22^2) + (7^2)

533 is the hypotenuse of four Pythagorean triples. Some of the triples have a greatest common factor greater than one, and the rest are primitive. Which are which?

• 92-525-533
• 117-520-533
• 205-492-533
• 308-435-533

Print the puzzles or type the solution on this excel file: 10 Factors 2015-06-22

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• 533 is a composite number.
• Prime factorization: 533 = 13 x 41
• The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 x 2 = 4. Therefore 533 has exactly 4 factors.
• Factors of 533: 1, 13, 41, 533
• Factor pairs: 533 = 1 x 533 or 13 x 41
• 533 has no square factors that allow its square root to be simplified. √533 ≈ 23.08679

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### 532 and Level 3

532 is the sum of consecutive prime numbers 263 and 269.

532 is also the 19th pentagonal number.

Print the puzzles or type the solution on this excel file: 10 Factors 2015-06-22

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• 532 is a composite number.
• Prime factorization: 532 = 2 x 2 x 7 x 19, which can be written 532 = (2^2) x 7 x 19
• The exponents in the prime factorization are 1, 2, and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1)(1 + 1) = 3 x 2 x 2 = 12. Therefore 532 has exactly 12 factors.
• Factors of 532: 1, 2, 4, 7, 14, 19, 28, 38, 76, 133, 266, 532
• Factor pairs: 532 = 1 x 532, 2 x 266, 4 x 133, 7 x 76, 14 x 38, or 19 x 28
• Taking the factor pair with the largest square number factor, we get √532 = (√4)(√133) = 2√133 ≈ 23.065125

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A Logical Approach to solve a FIND THE FACTORS puzzle: 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.