Euclid's Lemma

In Number Theory, the result that

A positive integer $p$ is a prime number if and only if $p|ab \Longrightarrow p|a$ or $p|b$

is attributed to Euclid

Proof of Euclid's lemma

There are two proofs of Euclid's lemma.

First Proof

By assumption $\gcd(a,b)=1$ , thus we can use Bezout's lemma to find integers $x,y$ such that $ax+by=1$ . Hence $c\cdot(ax+by)=c$ and $acx+bcy=c$ . Since $a\mid a$ and $a \mid bc$ (by hypothesis), we conclude that $a \mid acx + bcy =c$ as claimed.

Second Proof

We have $a\vert bc$ , so $bc=na$ , with $n$ an integer. Dividing both sides by $a$ , we have $\frac{bc}{a}=n$ . But $\gcd(a,b)=1$ implies $b/a$ is only an integer if $a=1$ . So $\frac{bc}{a} = b \frac{c}{a} = n$ , which means $a$ must divide $c$ .

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Euclid's Lemma

Proof of Euclid's lemma