Proof for Product of Congruences mod P Consider $Z_p$, with $p$ prime. Prove that $[x][x] = [y][y]$ if and only if $[x] =[y]$ or $[x] =[-y]$. 
I think this question comes down to the fact that $Z_p$ is cyclic but am not sure. 

Hint: One direction is easy, and does not require primality. 
From a number-theoretic point of view, the harder direction boils down to the fact that $p$ divides $x^2-y^2$ if and only if it divides $(x-y)(x+y)$.  
And if a prime divides a product, it divides (at least) one of the terms. 
