A Multiplication Based Logic Puzzle

865 and Level 1

Print the puzzles or type the solution on this excel file: 12 factors 864-874

865 is the sum of two squares two different ways:

  • 28² + 9² = 865
  • 24² + 17² = 865

865 is the hypotenuse of four Pythagorean triples, two of which are primitives:

  • 260-825-865, which is 5 times (52-165-173)
  • 287-816-865, which is 24² – 17², 2(24)(17), 24² + 17²
  • 504-703-865 which is 2(28)(9), 28² – 9², 28² – 9²
  • 519-692-865, which is (3-4-5) times 173

You could see 865’s factors in two of those Pythagorean triples, and here they are again:

  • 865 is a composite number.
  • Prime factorization: 865 = 5 × 173
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 865 has exactly 4 factors.
  • Factors of 865: 1, 5, 173, 865
  • Factor pairs: 865 = 1 × 865 or 5 × 173
  • 865 has no square factors that allow its square root to be simplified. √865 ≈ 29.41088
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Comments on: "865 and Level 1" (2)

  1. I like the way that although 865 does not have an integral square root it relates two pairs of integers, (28,9) and (24,17), by being the common hypotenuse of two right-angled triangles. The Pythagorean case when one of the four numbers thus related is zero collapses one of the triangles into just a line – much less beautiful?

    Liked by 1 person

    • I’m not sure what you meant when you wrote, “The Pythagorean case when one of the four numbers thus related is zero collapses one of the triangles into just a line – much less beautiful?” Are you saying that 865² + 0² = 865² is not as beautiful as 287² + 816² = 865²?

      Like

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