Why is one not prime or composite?

Answer 1

One is not composite because it has no factor other than 1 or itself.

In a sense 1 is prime, but the definition specifically excludes it from primality because allowing it to be a prime would mean that the factors of any number would include 1. Mathematicians prefer not to waste time or ink on something trivial.

Answer 2

1 cannot be a composite number because it is not divisible by any number other than itself.

It is not a prime because of the fundamental theorem of arithmetic: the unique prime factorisation theorem. This states that any positive integer greater than 1 can be expressed as a product of a unique set of primes. If 1 were considered a prime then you could add any number of 1s to the set of factors and the product would not change. For example,12 = 2*2*3 or 1*2*2*3 or 1*1*1*1*2*2*3 : so that the factorisation is no longer unique.