State/Primes dividing a product must divide a factor

 =Proof=

Products of two numbers
We begin by proving that if a prime divides a product of two natural numbers, it must divide one of the two numbers.

Products of many numbers
Now, we prove that primes which divide a product $\prod_{i=1}^k a_i$ of many numbers must divide a factor. We prove this by induction on the number of terms, $k$, in the product.