I have a question about congruency...
I understand that: $$ 12 \equiv 7 \bmod 5 $$ $$ \text {is equivalent to:} $$ $$ 5|12-7 $$ but this doesn't seem to hold for: $$ 2 \equiv 8 \bmod 6 $$ $$ \text {the conclusion i come to is:} $$ $$ 6|-6 $$ are these equivalent??
Yes, $6\mid-6$. That simply means that there is some integer $k$ such that $-6=6k$, and there is: take $k=-1$. There is no requirement that the integer be positive.