Divisor of Product may not be Divisor of Factors

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

Let:
 * $c \mathop \backslash a b$

where $\backslash$ expresses the relation of divisibility.

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

Proof
Proof by Counterexample:

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

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

But neither $6 \mathop \backslash 4$ nor $6 \mathop \backslash 3$.