Question

how can we prove that '1' is neither prime nor composite?

Answer 1

If 1 is prime, then it contradicts the Fundamental Theorem of Arithmetic. Ex. 10=1x2x5=1^100x2x5 which are 2 different factorizations of 10.
On the other hand, if 1 is composite, then it can be factored into some primes less than itself, but there are no primes less than 1.
Therefore, 1 is neither prime nor composite.