Show that if a prime number $p|a^n$ then $p|a$

Question

The title says it all, how can I prove the following:

Show that if a prime number $p|a^n$ then $p|a$

What do you already know, Euclid's Lemma, Bezout's GCD Identity, or uniqueness of prime factorizations? What have you tried and where are you stuck? — Bill Dubuque, Jan 14 '14 at 20:49

score 0 · Accepted Answer · answered Jan 14 '14 at 20:59

0

Hint: $p \mid m \, n \Longrightarrow p \mid m$ or $p \mid n$ for prime $p.$

answered Jan 14 '14 at 20:59

Leox

8,120

score 0 · Answer 2 · answered Jan 14 '14 at 21:00

0

What it means for $p$ to be prime: For any $a,b\in\mathbb{Z}$, if $p|(ab)$, then either $p|a$ or $p|b$. So what can you say if $p|a^2$? How does this help you with your problem?

answered Jan 14 '14 at 21:00

Neal

32,659

Show that if a prime number $p|a^n$ then $p|a$

2 Answers2