Past exam of the year 1 of the computer science course of the University of Oxford / 2015 / Discrete mathematics, continuous mathematics, and probability (resit) / 3 / c / Solution

By Euclidean division, write $a=qn+r$ and $b=sn+t$, with integers $q,s$ and $0\le r,t<n$. The hypothesis is $a\equiv b(\mathrm{mod}n)$. Apply the addition rule from part (b)(i) with $-qn\equiv -sn\equiv 0(\mathrm{mod}n)$ to obtainTherefore $n\mid r-t$. But $-n<r-t<n$, and the only multiple of $n$ in that interval is zero. Henceso the two remainders agree.