Is 3 Divisible by n if 3 is Divisible by n^2?

[cubicmonkey]

I'm completely lost on this one. I need this to be able to solve that the square root of three is irrational. So it's a proof within a proof, but I like this way best. Please help me out.
This is what I need to know how to prove.

3|n^2 implies 3|n where n is some integer.

 

[HallsofIvy]

3 is a prime number. If 3 were not a prime factor of n, it could not be a factor of n².

 

[cubicmonkey]

I think I understand what you're saying and that looks like a contrapositive proof, but you don't actually prove it. Could you elaborate?

 

[pedja]

I'm completely lost on this one. I need this to be able to solve that the square root of three is irrational. So it's a proof within a proof, but I like this way best. Please help me out.
This is what I need to know how to prove.

3|n^2 implies 3|n where n is some integer.

Suppose :

$$n \equiv a \pmod 3 ~\text{and}~ a \neq 0$$
then :
$$n^2 \equiv a^2 \pmod 3 ~\text{and}~ a^2 \neq 0$$
hence :
$$3 \nmid n^2$$
contradiction .

Q.E.D.

 

[20Tauri]

I think the Fundamental Theorem of Arithmetic (uniqueness up to order of prime factorizations) might help?