pythagorean triples

• November 23rd 2009, 02:25 PM
keityo
pythagorean triples
prove that if x,y,z is a Pythagorean triple, than at least one of x,y,z is divisible by 5.

What I have so far: let $x = r^2 - s^2, y = 2rs, z = r^2 + s^2$
Then try out all of the cases. If $5\nmid x \mbox{ and } 5\nmid y,\mbox{ then } 5\mid z$ and use congruences to show that it is true, for all of the possible values of r and s.

Is there a more elegant way to prove this, because what I have so far is too tedious.
• November 23rd 2009, 08:15 PM
CaptainBlack
Quote:

Originally Posted by keityo
prove that if x,y,z is a Pythagorean triple, than at least one of x,y,z is divisible by 5.

What I have so far: let $x = r^2 - s^2, y = 2rs, z = r^2 + s^2$
Then try out all of the cases. If $5\nmid x \mbox{ and } 5\nmid y,\mbox{ then } 5\mid z$ and use congruences to show that it is true, for all of the possible values of r and s.

Is there a more elegant way to prove this, because what I have so far is too tedious.

Consider the possible values of squares modulo $5$, then all the possible values of $x^2+y^2$ modulo $5$ and which of these can a square modulo 5.

CB