Does Euclid's Lemma generalize from primes to any integer?

Question

If p is prime and p|a₁a₂...a_n, then p|a_i for some i

I find it hard understanding what this lemma is saying, if p was any integer, the implication seems still valid. Thank everyone for having a look and I'd appreciate if you help me.

Here is an example in which in which this lemma is used, hope it helps:

Show that if p is prime, the only solutions of x² ≡ 1 (mod p) are integers x such that x ≡ 1 (mod p) or x ≡ −1 (mod p).

Solution: Suppose that x² ≡ 1 (mod p). Then p divides x² −1 = (x+1)(x−1). By the Lemma above it follows that p|(x+1) or p|(x−1), so x ≡ −1 (mod p) or x ≡ 1 (mod p)