# Euclid's Lemma for Prime Divisors/Corollary

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## Corollary to Euclid's Lemma for Prime Divisors

Let $p, p_1, p_2, \ldots, p_n$ be primes such that:

- $p \divides \ds \prod_{i \mathop = 1}^n p_i$

Then:

- $\exists i \in \closedint 1 n: p = p_i$

## Proof

From Euclid's Lemma for Prime Divisors: General Result, $p \divides p_i$ for some $i$.

But by the definition of a prime number, the only divisors of $p_i$ are $1$ and $p_i$ itself.

As $1$ is not prime, it follows that $p = p_i$.

Hence the result.

$\blacksquare$

## Source of Name

This entry was named for Euclid.