I hope you guys are trolling...but in case you're actually thinking about carrying on with this
line of reasoning, I thought I'd point out that you only need to consider prime n since if n is
composite, say n = pq, you have x^(pq) = (x^p)^q = (x^q)^p, so if a solution existed for n, it would also
exist for p and for q.

And no, you cannot prove FLT generally using simple congruences...but you can definitely prove it for
a bunch of small primes.