Common Divisor in Integral Domain Divides Linear Combination

Theorem
Let $$c$$ be a common divisor of two integers $$a$$ and $$b$$.

That is: $$a, b, c \in \mathbb{Z}: c \backslash a \land c \backslash b$$.

Then $$c$$ divides any integer combination of $$a$$ and $$b$$:

$$\forall p, q \in \mathbb{Z}: c \backslash \left({p a + q b}\right)$$