The quotient remainder theorem

When we want to prove some properties about modular arithmetic we often make use of the quotient remainder theorem.
It is a simple idea that comes directly from long division.
The quotient remainder theorem says:
Given any integer A, and a positive integer B, there exist unique integers Q and R such that
A= B * Q + R where 0 ≤ R < B
We can see that this comes directly from long division. When we divide A by B in long division, Q is the quotient and R is the remainder.
If we can write a number in this form then A mod B = R

Examples

A = 7, B = 2
7 = 2 * 3 + 1
7 mod 2 = 1
A = 8, B = 4
8 = 4 * 2 + 0
8 mod 4 = 0
A = 13, B = 5
13 = 5 * 2 + 3
13 mod 5 = 3
A = -16, B = 26
-16 = 26 * -1 + 10
-16 mod 26 = 10