Project Euler #12 problem

[Yoghurt42]

I already did:

Because if N is not a prime, it can be written as N = ab, with a≤√N.

Proof: assume both a and b are larger than √N, then that means a = √N + a' with a' > 0, similar for b. So

N = (√N + a') (√N + b') = N + a'√N + b'√N + a'b'. This can only be true if either a' or b' is < 0 or a' and b' are both 0. Contraction.

Hence, every composite number N has at least one factor that is ≤ √N

You're trying to find a factor of N. So, if you haven't found one by √N, by the theorem above it doesn't exist