Divisor Divides Multiple/Proof 1

Theorem

Let $a, b$ be integers.

Let:

$a \divides b$

where $\divides$ denotes divisibility.

Then:

$\forall c \in \Z: a \divides b c$

Proof

Let $a \divides b$.

From Integer Divides Zero:

$a \divides 0$

Thus $a$ is a common divisor of $b$ and $0$.

From Common Divisor Divides Integer Combination:

$\forall p, q \in \Z: a \divides \paren {p \cdot b + q \cdot 0}$

Putting $p = c$ and $q = 1$ (for example):

$a \divides \paren {c b + 0}$

Hence the result.

$\blacksquare$