Division Theorem/Positive Divisor/Positive Dividend/Existence/Proof 2

Proof
Let $a = 0$.

It is immediately apparent that $q = r = 0$ is the only possible solution with $0 \le r < b$.

Let $a > 0$ and $b = 1$.

Then from the condition $0 \le r < b$ it follows that $r = 0$ and hence $q = a$.

Let $a > 0$ and $b > 1$.

By the Basis Representation Theorem, $a$ has a unique representation to the base $b$: