greatest common factor

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

534 and Level 5

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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?

534 Puzzle

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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534 Logic

482 and Level 2

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Can you determine the greatest common factor of the three numbers in this Pythagorean triple: 240-418-482? Hint: it is one of the factors of 482 listed below the puzzle.

482 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-05-04

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  • 482 is a composite number.
  • Prime factorization: 482 = 2 x 241
  • 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 482 has exactly 4 factors.
  • Factors of 482: 1, 2, 241, 482
  • Factor pairs: 482 = 1 x 482 or 2 x 241
  • 482 has no square factors that allow its square root to be simplified. √482 ≈ 21.954498

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 482 Factors

470 Greatest Common Factors of Pythagorean Triples.

/ ivasallay / 6 Comments
  • 470 is a composite number.
  • Prime factorization: 470 = 2 x 5 x 47
  • 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 470 has exactly 8 factors.
  • Factors of 470: 1, 2, 5, 10, 47, 94, 235, 470
  • Factor pairs: 470 = 1 x 470, 2 x 235, 5 x 94, or 10 x 47
  • 470 has no square factors that allow its square root to be simplified. √470 ≈ 21.67948

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470 is the hypotenuse of the non-primitive Pythagorean triple 282-376-470. What is the greatest common factor of those three numbers?

The greatest common factor will always be a factor of the smallest of the three numbers, but it will also be a factor of the smallest difference between the three numbers as well. Let’s find those differences. Note: the difference between the 282 and 470 will not be the smallest difference so there is no need to find that one. We only need to find these two differences:

470 difference

In the case of this Pythagorean triple the differences are equal to each other which means that the difference, 94*, is also the greatest common factor of the three numbers! Go ahead and try dividing each number in the triple by 94. You will discover that this Pythagorean triple is just 3-4-5 multiplied by 94.

*This statement is only true of Pythagorean triples. For example the following numbers also have differences of 94, but the greatest common factor is not 94, but a factor of 94:

  1. The greatest common factor of 283-377-471 is 1.
  2. The greatest common factor of 284-378-472 is 2
  3. The greatest common factor of 329-423-517 is 47

Mathchat has written an excellent post on finding the greatest common factor of three or more numbers that can be used for all integers in general.

But as far as Pythagorean triples are concerned, anytime the corresponding differences of a Pythagorean triple are equal to each other, then that Pythagorean triple is just 3-4-5 multiplied by the difference. There are an infinite number of such triples, and 282-376-470 is just one of them.

3-4-5 Pythagorean Triple Sequence

Now remember there is an infinite number of primitive Pythagorean triples, and every one of those triples can be multiplied by each of the infinitely many counting numbers. A graphic like the one above could be made for every primitive triple followed by each of its multiples. For example 5-12-13, 10-24-26, 15- 36-39, etc. would be another infinite series of Pythagorean triples.

You could say the total number of Pythagorean triples equals infinity times infinity!

469 and Level 4

/ ivasallay / 4 Comments

469 is the short leg in the Pythagorean triple 469-1608-1675. What is the greatest common factor of those three numbers? Hint: Don’t let the larger numbers scare you; the greatest common factor is a factor of 469, the smallest of those three numbers, and its factors are listed below the puzzle.

469 Puzzle

Print the puzzles or type the solution on this excel file:  12 Factors 2015-04-20

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  • 469 is a composite number.
  • Prime factorization: 469 = 7 x 67
  • 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 469 has exactly 4 factors.
  • Factors of 469: 1, 7, 67, 469
  • Factor pairs: 469 = 1 x 469 or 7 x 67
  • 469 has no square factors that allow its square root to be simplified. √469 ≈ 21.6564

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469 Logic