MultProof

From $a\equiv b\pmod m$ and $a\equiv b\pmod n$, we can write $a=ms+b$ and $a=nt+b$ for some $s$ and $t$. Here we have $ms=nt$. Assume that $m$ and $n$ are coprime. Then there are $x$ and $y$ such that $mx+ny=1$. Therefore, $s=smx+sny=ntx+sny$, and we have $a=m(ntx+sny)+b=mn(tx+sy)+b\equiv b\pmod{mn}$. $\Box$