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.