Does Euclid's Lemma apply to squared numbers?

Question

Euclid's lemma states that: if a prime $p$ divides the product $ab$ of two integers $a$ and $b$, then $p$ must divide at least one of those integers $a$ and $b$.

My question is can $a=b$? or must they be different integers?

If they can would I be able to use Euclid's Lemma to prove a question such as this:

Prove that if $n^2$ is divisible by $13$ then $n$ is divisible by $13$

Answer 1

Of Course you can set $a=b$ and get the relation that if $p|a^2 $ then $p|a$

Proof : Although trivial , consider the prime factorization of $a$ :

$$a = p_1^{a_1}*p_2^{a_2}*p_3^{a_3}....p_n^{a_n}$$

so the prime factorization of $a^2$ would be: $$a^2 = p_1^{2a_1}*p_2^{2a_2}*p_3^{2a_3}....p_n^{2a_n}$$

Now since all the primes that occurs in $a^2$ also occurs in $a$ , if $p_k$ is in $a^2$ then $p_k$ should also be in $a$.

Hence : $p|a^2$ $\implies$ $p|a$

Answer 2

The short answer is "Yes". If $n^2=13\times13$, then $n=13$ so $13$ divides $n$.