I am currently confused on a problem in a practice set I have been given. It reads the following:
Let $d$ be a positive integer and let $n$ be an integer. Prove that $d | n$ if and only if in the unique expression $n = dq + r$ with $q,r \in \mathbb{Z}$ and $0\leq r < d$ the remainder is $r = 0$.
To me the question seems totally trivial in the sense that it follows from the definition of the notation $x|y$ that $y=m\bullet x$ for some $x\in\mathbb{Z}$ and in the other direction: if one assumes that $n=dq+r$ with $r=0$ then $q$ is by definition an integer such that $n$ can be expressed by the multiplication of $d$ (the divisor) and $q$ and so obviously $d|n$.
At first I thought this was asking to prove the existence of the division algorithm, but upon reflection it seems a totally different question. Could anyone provide some elucidation?