In my lecture notes it is written that $a \equiv b \mod n$ with $a, b, n \in \Bbb Z$ could also be understood in the sense that $a$ divided by $n$ gives $b$ as a remainder. I don't see why this should be true though.
Take $a = 5, \ b = 11$. Obviously $5 \equiv 11 \mod 3$, but $5$ divided by $3$ does not give $11$ as a remainder. What am I missing here?