Divisor of Product may not be Divisor of Factors

Theorem

Let $a, b, c \in \Z_{>0}$ be (strictly) positive integers.

Let:

$c \divides a b$

where $\divides$ expresses the relation of divisibility.

Then it is not necessarily the case that either $c \divides a$ or $c \divides b$.

Proof

Proof by Counterexample:

Let $c = 6, a = 3, b = 4$.

Then $6 \times 2 = 12$ so $c \divides a b$.

But neither $6 \divides 4$ nor $6 \divides 3$.

$\blacksquare$

Sources

• 1982: Martin Davis: Computability and Unsolvability (2nd ed.) ... (previous) ... (next): Appendix $1$: Some Results from the Elementary Theory of Numbers