Both $5$ and $13$ are prime numbers. Isn't that enough?

I know that! But how to prove $5^n$ is Coprime with 13

Begin with: “If $p$ is prime and $p$ divides both $5^n$ and $13$, then…”

So are the comments helpful, or do you need more?

Euclid's Lemma tells us that $13\,|\,5^n\implies 13\,|\,5$. A contradiction.

@Cornman, if p|x is true, does that mean p and y are Coprime?

I have no idea about how you jumped to that conclusion. Care to explain?

@JoséCarlosSantos, If p|13 and p|$5^n$ then p|13*$5^n$ ?

Sure. But then how do you get that $p\mid13^n$?

I dont't think you understood my problem. My problem is how to proove gcd(13, $5^n$) = 1

What is the main result you seek to prove? It seems you may be using a roundabout method.

I want to prove that $5^n$ and 13 are coprime

But you seem to want to use that as as Lemma for something else "I have proved that ...." whose proof may not require that Lemma. If not, why did you "prove that"?

@BillDubuque Yes Gauss Lemma but I didnt know how

Throw away the math for a minute, and use your intuition. Two positive integers $a,b$, each greater than $1$, will not be coprime if and only if there is some prime number $p$ that occurs in the prime factorizations of each of $a$ and $b$.

Why did you prove that $5^{n+4}-5^n \equiv 0\pmod{13}?$ Is that part of some bigger problem? Did you mean that you are trying to prove that, by using that $5$ and $13$ are coprime, using Euclid's Lemma (not Gauss's Lemma - that name is used only in France)

@BillDubuque my native language is French! Yes its a part of a bigger problem.

@user2661923 Thank youu!

Your recent posts are confusing due to many English errors, e.g. you wrote "I proved" when you mean "I need to prove" and $\,n\,$ divides $\,13\,$ when you meant $\,13\,$ divides $\,n.\,$ This makes it difficult to determine what you intend. Please be more careful composing your posts,

Is your main goal to prove that $5^{n+4}-5^n = 13k\,$ for some integer $k$ or is your goal to prove that $\,5^4-1 = 13k\,$ for some integer $k?\ \ $

@BillDubuque I could be mistaken, but I think that his main goal was to determine whether $5^n$ and $13$ are coprime. He was clutching at straws, when (apparently) he proved that since $(5^4 -1) \equiv 0 \pmod{13}$, then $5^{n+4} - 5^n \equiv 0\pmod{13}.$ Personally, I see no way that this relates to the question of whether $5^n$ and $13$ are coprime, which is why I inferred that the OP was clutching at straws.

@BillDubuque, No! I actually meant that "I proved"

@user2661923, Yes you are actually correct xD

So you were trying to prove that $5^n$ is coprime to $13$ by induction on $n$, using $5^{n+4} \equiv 5^n\pmod{13}$ as the descent step? This is certainly possible, and if that is what you intended then you should edit the question to clarify that (and unaccept the current answer in order to get one that pertains to the intended question).