Prove that if x2 y2 z2 then one of x and y is a multiple o

Prove that if x^2 + y^2 = z^2, then one of x and y is a multiple of 3.

Solution

We prove by contradiction

Assume, x and y are not multiples of 3

So, x,y are 1 or -1 mod 3

So, x^2=y^2=1 mod 3

x^2+y^2=1+1=2 mod 3

So, z^2=2 mod 3

But, 2 is not a quadratic residue modulo 3

So a contradiction

Hence, one of x and y is a multiple of 3