Given a statement like $b$ divides $a$ means there exists some integer $k$ such that $a = b\cdot k$, does that mean if $b$ does not divide $a$, then there are integers $q$ and $r$ such that $a = q\cdot b +r$ for $0<r<b$ according to the division algorithm?
Am I assuming too much here?
EDIT: Let me also clarify that I am assuming $a,b\in\mathbb{Z}$ and both are positive.
Best Answer
Yes. There's a simpler representation.
To say $b$ does not divide $a$ means that $b$ is not a factor of $a$. That is, if you try to do the division, you'll have leftovers - the remainder $r$.