Exercise 1488Number theory › Divisibility and gcdL1Added 2 Nov 2026

A prime divisor property

#primes#divisibility#Euclid's lemma

Statement

Let $p$ be prime and $a,b$ integers. Show that if $p\mid ab$ , then $p\mid a$ or $p\mid b$ .

Hint

If $p\nmid a$ , then $\divisibility(p,a)=1$ . Apply Gauss's lemma.

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Solution

Case 1. If $p\mid a$ , done.

Case 2. If $p\nmid a$ , then since $p$ is prime, $\divisibility(p,a)=1$ . By Gauss's lemma (if $\divisibility(p,a)=1$ and $p\mid ab$ , then $p\mid b$ ), we get $p\mid b$ .

Remark. This property characterizes prime numbers in $\mathbb{Z}$ and is the key to unique factorization.

A prime divisor property

Statement

Solution

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