Possible to find in general a $c$ such that for two specific integers,$a$ and $b$, $a \equiv b \bmod c$?

Question

A professor of mine gave us a question to think about regarding hashing and I have an idea, but it hinges on this being possible and I'm not too sharp on modular arithmetic. To give an example, let's say $a = 243$ and $b = 1724$. Can we find a $c$ that satisfies $243\equiv 1724 \bmod c$? And will this solution method work for any numbers $a$ and $b$?

Thanks.

Answer 1

In $\mathbb{Z}/c\mathbb{Z}$ we have the equation $243=1724$. So we have $0=1724-243=1481$. Then $c=1481$ works. In general, $a=b$ implies that $a-b=0$ in $\mathbb{Z}/c\mathbb{Z}$, i.e., $c\mid a-b$.