Divisibility Problem: How can I solve this?

Question

Suppose that $a,b,q,r$ are any integers such that $b > 0$ and $a = bq + r$, with $0\le r<b$, and suppose $b|a$.

Must it be the case that $r = 0$? Justify your answer.

Can anyone please let me know how can I start this problem ? Thanks

Answer 1

Hint: since $b|a$, this means that (by definition) $a=k\cdot b$ for some $k\in\mathbb Z$. This means that $$qb+r=kb.$$

All you have to prove now is that $q=k$.

Hint 2: The equation is equivalent to $r=(k-q)\cdot b$.

Answer 2

To elaborate on my comment, since both $a$ (by hypothesis) and $bq$ (obviously) are multiples of $b$, $r = a-bq$ is also a multiple of $b$. How many multiples of $b$ are there in $\{0,\dots,b-1\}$?