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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.

makansij
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    You can’t just say it is $nm$, because this may be equal to or larger than $y$. For example, the remainder of dividing $5$ by $3$ is $2$, and the remainder of dividing $11$ by $3$ is $2$, but the remainder of dividing $55$ by $3$ is not $4$. – Arturo Magidin Sep 02 '19 at 22:48
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    Not quite: it is only the remainder of $nm$ modulo $y$. – Bernard Sep 02 '19 at 22:49

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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 $y$).