Equivalence Relation Proof for modular arithmatic Given this modular relation:
$x^3 \equiv y \pmod{3}$
how would you go about proving the transitivity of the system? I have proven the reflexivity, and symmetry pretty easily but the transitivity is giving me many problems, and I feel like im not setting up the problem correctly.

It is a more general result that the congruence modulo an integer $p$ relation; $\equiv$; is an equivalence relation on the set of integers. That is, we say that two integers $m$ and $n$ are equivalent modulo $p$ if $p$ divides $m-n$
$$m\equiv n (\mod p)\Leftrightarrow p | (m-n)$$
Pour la transitivité: supposons que $m\equiv n (\mod p)$ and $n\equiv q (\mod p)$
then 
$p|m-n$ and $p|q-n$ hence $p|(m-n)-(q-n)=m-q$ This is because if an integer divides two other integers then it divides any linear combination of them, in particular it divides their difference.
So $p|m-q$ which means that $m\equiv q (\mod p)$ hence the relation "$\equiv$ modulo $p$" is transitive.
