A Multiplication Based Logic Puzzle

Because both of its prime factors are hypotenuses of primitive Pythagorean triples, 685 is the hypotenuse of FOUR Pythagorean triples. Two are primitive; two are not:

  • 37-684-685 which was calculated from 19² – 18², 2(19)(18), 19² + 18²
  • 156-667-685 which was calculated from 2(26)(3), 26² – 3², 26² + 3²
  • 411-548-685 (What factor of 685 is the greatest common factor of those 3 numbers?)
  • 440-525-685 (and what is their greatest common factor?)

As you may have notice from those calculations, 685 is the sum of two squares two different ways:

  • 19² + 18² = 685
  • 26² + 3² = 685

685 is the 19th centered square number because 18 and 19 are consecutive numbers and 19² + 18² = 685. There are 685 small squares of various colors in this graphic.

685 is the 19th Centered Squared Number

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  • 685 is a composite number.
  • Prime factorization: 685 = 5 x 137
  • 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 685 has exactly 4 factors.
  • Factors of 685: 1, 5, 137, 685
  • Factor pairs: 685 = 1 x 685 or 5 x 137
  • 685 has no square factors that allow its square root to be simplified. √685 ≈ 26.172505.

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