Dumb question about the division algorithm.

Question

The theorem about the division algorithm says:

Given a, b $ \in \mathbb{Z}, b \neq{0}, $ there exist unique numbers q and r , $q,r \in \mathbb Z $such that $ a = bq + r , 0 \lt r \lt |b| $.

Can q be zero?

Answer 1

Yes, definitely. Let $a = 4, b = 12$.
Then $4 = 12\times 0 + 4$, and $0 < 4 < 12$.

You can see that this happens in general. Whenever $|b| > a > 0$, we have $a = b \times 0 + a$, so that $r = a$. And since $|b| > a$ (by assumption), we have $0 < r < |b|$, as required.