Definition:Remainder

Let $$a, b \in \mathbb{Z}$$.

From the Division Theorem, we have that:

$$\forall a, b \in \mathbb{Z}, b \ne 0: \exists! q, r \in \mathbb{Z}: a = q b + r, 0 \le r < \left|{b}\right|$$

The value $$r$$ is defined as the principal remainder of $$a$$ on division by $$b$$.