485 is the hypotenuse of four Pythagorean triples. Which ones are primitive and which ones aren’t?

- 44-483-485
- 93-476-485
- 291-388-485
- 325-360-485

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

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- 485 is a composite number.
- Prime factorization: 485 = 5 x 97
- 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 485 has exactly 4 factors.
- Factors of 485: 1, 5, 97, 485
- Factor pairs: 485 = 1 x 485 or 5 x 97
- 485 has no square factors that allow its square root to be simplified. √485 ≈ 22.0227155

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This one is for me, right?

Okay, 44-483-485 doesn’t seem like it’s even worth playing with: primitive for sure.

93-476-485 likely primitive as well ( can’t fathom that 476 and 485 have any common factors).

325-360-485, well, are all multiples of 5, so it won’t be primitive (how am I doing?)

291-388-485: now this is the interesting one. seems 291 doesn’t have many factors, but it divides it by 3, and you get a 97, which, if you’ve done your math right in this post, also goes neatly into 485 (thank you) so 291-388-485 is not primitive!

What confounds me most about all this is that 485 is the hypotenuse of four Pythagorean triples. It just never occurred to me that a number could be the hypotenuse of more than one Pythagorean triplet.

Thank you for recognizing that this problem was just for you! You got everything right, too!

If the hypotenuse of a Pythagorean triple is a prime number, then it will not be the hypotenuse of any other Pythagorean triple. However, if the hypotenuse is a composite number, it might be the hypotenuse of other triplets as well.