Past exam of the year 1 of the computer science course of the University of Oxford / 2021 / Discrete mathematics, continuous mathematics, and probability / 1 / c / iii / Solution

Take $(a,b,c,n)=(1,4,2,6)$. These are four distinct positive integers and $n\ge 3$. Their products are $ac=2$ and $bc=8$, whose difference is $6$, so $ac\equiv bc(\mathrm{mod}6)$. Here the correct reduced modulus is $6/\gcd (2,6)=3$, and indeed $1\equiv 4(\mathrm{mod}3)$. The example illustrates why cancelling a noninvertible factor may change the modulus in modular arithmetic.