Cancellation

Proposition 2.1   If $\gcd(c,n)=1$ and

$$ac\equiv bc\pmod{n}$$

then $a \equiv b\pmod{n}$.

Proof. By definition

$$n \mid ac - bc = (a-b)c.$$

Since $\gcd(n,c)=1$, it follows that $n\mid a-b$, so

$$a \equiv b\pmod{n},$$

as claimed. $\qedsymbol$