Suppose divides but . Since is prime, we then have . By Bézout’s identity there exist with . Multiplying both sides by gives

Since divides and also the product , both summands above are multiples of , so divides . The other case is similar.

The same reasoning extends by induction on to more factors: if divides , then divides for some . Primality is essential: the statement is false if is not prime, since for example , yet and .