Is each integer $n > 2$ is divisible by $4$ or by an odd prime number While reading through the proof of Fermat's Last Theorem, I came across this statement. "Each integer $n > 2$ is divisible by $4$ or by an odd prime number"
But I don't know how to prove it. 

claim: Every integer $n$ at least $3$ is divisible either by $4$ or by an odd prime.
$3$ possible cases:

*

*The prime factorisation of $n$ contains exactly at least two ‘$2$’s.
So, $n$ is divisible by $4.$

*The prime factorisation of $n$ contains exactly one ‘$2$’, in which
case—since $n\geq3$—it must contain an odd prime.
So, $n$ is divisible by an odd prime.

*The prime factorisation of $n$ contains no ‘$2$’, in which case it
must contain an odd prime. So, $n$ is divisible by an odd prime.

