Pythagorean Triple

569 and Level 5

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When we divide the last two digits of prime number 569 by 4, we get a remainder of one. That means that 569 is the sum of two square numbers, specifically, 20² + 13² = 569.

From those two square numbers we will obtain 569 as the hypotenuse of the primitive Pythagorean triple 231-520-569:

  • 20² – 13² = 231
  • 2 x 13 x 20 = 520
  • 20² + 13² = 569

569 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-07-27

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  • 569 is a prime number.
  • Prime factorization: 569 is prime.
  • The exponent of prime number 569 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 569 has exactly 2 factors.
  • Factors of 569: 1, 569
  • Factor pairs: 569 = 1 x 569
  • 569 has no square factors that allow its square root to be simplified. √569 ≈ 23.85372

How do we know that 569 is a prime number? If 569 were not a prime number, then it would be divisible by at least one prime number less than or equal to √569 ≈ 23.8. Since 569 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 569 is a prime number.

569 Logic

562 and Level 4

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562 is the sum of all the prime numbers from 23 to 71.

562 is also the hypotenuse of the Pythagorean triple 320-462-562. Which factor of 562 is the greatest common factor of those three numbers?

562 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-07-20

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  • 562 is a composite number.
  • Prime factorization: 562 = 2 x 281
  • 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 562 has exactly 4 factors.
  • Factors of 562: 1, 2, 281, 562
  • Factor pairs: 562 = 1 x 562 or 2 x 281
  • 562 has no square factors that allow its square root to be simplified. √562 ≈ 23.7065

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562 Logic

560 and Level 3

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560 is the hypotenuse of the Pythagorean triple 336-448-560. What is the greatest common factor of those three numbers?

560 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-07-20

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  • 560 is a composite number.
  • Prime factorization: 560 = 2 x 2 x 2 x 2 x 5 x 7, which can be written 560 = (2^4) x 5 x 7
  • The exponents in the prime factorization are 4, 1 and 1. Adding one to each and multiplying we get (4 + 1)(1 + 1)(1 + 1) = 5 x 2 x 2 = 20. Therefore 560 has exactly 20 factors.
  • Factors of 560: 1, 2, 4, 5, 7, 8, 10, 14, 16, 20, 28, 35, 40, 56, 70, 80, 112, 140, 280, 560
  • Factor pairs: 560 = 1 x 560, 2 x 280, 4 x 140, 5 x 112, 7 x 80, 8 x 70, 10 x 56, 14 x 40, 16 x 35 or 20 x 28
  • Taking the factor pair with the largest square number factor, we get √560 = (√16)(√35) = 4√35 ≈ 23.664319

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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.

560 Factors

 

559 and Level 2

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All the prime numbers from 67 to 97 add up to 559, and all the prime numbers from 103 to 127 also add up to 559.

559 is also the hypotenuse of Pythagorean triple 215-516-559. What is the greatest common factor of those three numbers?

559 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-07-20

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  • 559 is a composite number.
  • Prime factorization: 559 = 13 x 43
  • 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 559 has exactly 4 factors.
  • Factors of 559: 1, 13, 43, 559
  • Factor pairs: 559 = 1 x 559 or 13 x 43
  • 559 has no square factors that allow its square root to be simplified. √559 ≈ 23.64318

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559 Factors

555 and Level 5

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555  is the hypotenuse of four Pythagorean triples:

  • 171-528-555
  • 180-525-555
  • 312-459-555
  • 333-444-555

Each of those Pythagorean triples has a greatest common factor. Can you figure out what each one is? Hint: each one will be a factor of 555 listed below the puzzle.

555 Puzzle

 

Print the puzzles or type the solution on this excel file: 12 Factors 2015-07-13

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  • 555 is a composite number.
  • Prime factorization: 555 = 3 x 5 x 37
  • 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 555 has exactly 8 factors.
  • Factors of 555: 1, 3, 5, 15, 37, 111, 185, 555
  • Factor pairs: 555 = 1 x 555, 3 x 185, 5 x 111, or 15 x 37
  • 555 has no square factors that allow its square root to be simplified. √555 ≈ 23.5584

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555 Logic

554 and Level 4

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554 is the hypotenuse of the Pythagorean triple 230-504-554. What is the greatest common factor of those three numbers?554 Puzzle

 

Print the puzzles or type the solution on this excel file: 12 Factors 2015-07-13

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  • 554 is a composite number.
  • Prime factorization: 554 = 2 x 277
  • 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 554 has exactly 4 factors.
  • Factors of 554: 1, 2, 277, 554
  • Factor pairs: 554 = 1 x 554 or 2 x 277
  • 554 has no square factors that allow its square root to be simplified. √554 ≈ 23.5372

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554 Logic

549 and Level 6

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549 is the hypotenuse of the Pythagorean triple 99-540-549. What is the greatest common factor of those three numbers?

549 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-07-06

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  • 549 is a composite number.
  • Prime factorization: 549 = 3 x 3 x 61, which can be written 549 = (3^2) x 61
  • The exponents in the prime factorization are 2 and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1) = 3 x 2  = 6. Therefore 549 has exactly 6 factors.
  • Factors of 549: 1, 3, 9, 61, 183, 549
  • Factor pairs: 549 = 1 x 549, 3 x 183, or 9 x 61
  • Taking the factor pair with the largest square number factor, we get √549 = (√9)(√61) = 3√61 ≈ 23.430749

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549 Logic

544 and Level 1

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544 is the hypotenuse of Pythagorean triple 256-480-544. Can you find the greatest common factor of those three numbers?

544 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-07-06

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  • 544 is a composite number.
  • Prime factorization: 544 = 2 x 2 x 2 x 2 x 2 x 17, which can be written 544 = (2^5) x 17
  • The exponents in the prime factorization are 5 and 1. Adding one to each and multiplying we get (5 + 1)(1 + 1) = 6 x 2 = 12. Therefore 544 has exactly 12 factors.
  • Factors of 544: 1, 2, 4, 8, 16, 17, 32, 34, 68, 136, 272, 544
  • Factor pairs: 544 = 1 x 544, 2 x 272, 4 x 136, 8 x 68, 16 x 34, or 17 x 32
  • Taking the factor pair with the largest square number factor, we get √544 = (√16)(√34) = 4√34 ≈ 23.3238

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544 Factors

 

538 and Level 2

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

538 Puzzle

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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538 Factors

535 and Level 6

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

535 Puzzle

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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535 Logic