How to prove there is at most one remainder then there is at least one remainder

Question

In equation $b=aq_1 + r_1$ and $b=aq_2 + r_2,\;\;$ I have proof $r_1 = r_2$ (there is at most one remainder). But how to proof there is at least one remainder?

Let $S = \{s\in \mathbb Z, s \geq 0 : \exists q \in \mathbb Z \text{ such that } b = aq + s\}$

Answer 1

The set of positive integers in the progression $b-qa$ for $q=0,1,...$ has a least element $r^*$ which is the remainder. The corresponding $q^*$ is the quotient.