perplexus.info :: Just Math : OOO3= one out of three

OOO3= one out of three (Posted on 2010-11-04)

Given an equation x^2+y^2+z^2=2010.
1. Prove that in every triplet of integers satisfying the above equation one number has to be even and two others odd.
2. How many integer solutions are there ?

Warning: 1 is very easy, 2 is quite tricky.

No Solution Yet Submitted by Ady TZIDON

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Puzzle Answer

| Comment 9 of 10 |

Reducing the equation Mod(4), we have:

x^2+y^2+z^2==2mod 4

This is possible only if two of x,^2 y^2, z^2 have the form 1(mod 4) , and the remaining one as: 0mod(4)

Hence, two of the triplets must be odd, while the remaining one is even.

Posted by K Sengupta on 2024-06-03 09:37:26

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