Determine remainder of the product of two numbers, when we know the remainder of each individually?

Question

I want to confirm some of my logic with remainders. If I know the remainder of $\frac{x}{y}$ is $n$, and the remainder of $ \frac{z}{y}$ is $m$, then can I say that the remainder of $ \frac{xz}{y}$ is $n*m$?

I don't have a formal proof, but I can't find a counter-example.

Answer 1

Hint: $x = qy+n$ and $z=ky+m$ for integers $q$ and $k$. So $xz = (qky+qm+nk)y+nm$. So...(spoiler alert: $nm$ might not necessarily be smaller than $y$).