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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re: Puzzle Answer

Comment 10 of 10 |

(In reply to Puzzle Answer by K Sengupta)

If only positive integers are considered, then there are precisely 8 non trivial triplets as follows: (1,28,35), (4,25,37), (5,7,44), (5,31,32) , (7,19,40), (11,17,40), (16,23,35) and (19,25,32).

Each of these can be permuted 6 ways, producing 48 positive integral solutions.

Each of x, y and z can, independently, be made positive or negative, multiplying the solutions by 2^3 = 8, making the final number of integer solutions 8*6*8 = 384.

Edited on June 3, 2024, 9:45 am

Posted by K Sengupta on 2024-06-03 09:44:10

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