Prove that $kn+r$ isn't divisible by $k$ if $r$ isn't divisible by $k$ and $k$,$n$ and $r$ are integers.

Question

I understand that this is trivial, but I never understood why, nor did I see a proof for it.

Answer 1

Prove the contrapositive. If $k\mid r$, then $r=kj$ for some integer $j$ whence $$ kn+r=k(n+j) $$ as desired.

Answer 2

1. Suppose $k$, $n$, and $r$ are integers and set $A = kn+r$.
2. If $r$ is an integer multiple of $k$ (say, $r = ka$) then we can factor: $$A = kn+ r = kn +ka = k(n+a),$$ which means that $A$ is a multiple of $k$.
3. Therefore, if $r$ is a multiple of $k$ then $A$ is.
4. The contrapositive of this statement is that if $A$ is not a multiple of $k$, then $r$ isn't either.
5. That's what we wanted to show, so we're done.