A difference in a formula of theorem 4.2(e) on congruence relations.

Question

The statement of the theorem said :

If $a \equiv b \pmod n$ then $ac \equiv bc \pmod n$.

But I have seen it in other place as:

$a \equiv b \pmod n$ then $ac \equiv bc \pmod {nc}$.

Are they equivalent ? if so why?

Answer 1

Well, they both hold.

By definition of modular congruence, $a\equiv b\pmod n$ means $n|b-a$, that is, $b-a=dn$ for some $d$.
But then $bc-ac=dnc$, showing $ac\equiv bc\pmod{nc}$.

Generally, if $x\equiv y\pmod{nc}$, we always have $x\equiv y\pmod n$, too, because the condition means $nc|y-x$ which implies $n|y-x$.