Why does the following equivalence hold true? $m$ and $n$ are natural numbers. (This is part of a proof)
$ 123^m −33^n \equiv 3^m −3^n \pmod{10} \equiv 3^m(3^{n−m} −1) \pmod{10} $
I understand that $ 3^m3^{n−m}=3^n $ and $-1(3^m)=-3^m$, but I don't understand why $3^m-3^n \equiv 3^n - 3^m\pmod {10}$.
hint
$$123\equiv 3 \mod 10\;\implies$$ $$123^m\equiv 3^m \mod 10$$
If $a\equiv b \mod c$
and $A\equiv B \mod c$
then $a-A\equiv b-B \mod c$.
But the statement you did not understand is false with $(m=2;n=1)$. $$3^2-3\equiv 6\mod 10$$ but $$3-3^2=-6\equiv 4 \mod 10$$