Common Divisor Divides Integer Combination/Corollary

Corollary to Common Divisor Divides Integer Combination

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

That is:

$a, b, c \in \Z: c \divides a \land c \divides b$

Then:

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

Converse does not Hold

Let $a, b, c \in \Z$ be integers.

Let:

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

Then it is not necessarily the case that:

$c \divides a \land c \divides b$

Proof

From Common Divisor Divides Integer Combination:

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

The result follows by setting $p = q = 1$.

$\blacksquare$

Sources

• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 11.4$: The division algorithm