Why are zero and one neither prime nor composite?

Answer 1

The definition of a Prime number is a natural number that has exactly two distinct natural number divisors. And the definition of a composite number is a natural number that has more than two finite, distinct natural number divisors.

The number one has just one natural number divisor (1), thus it doesn't fit in either category. Zero has an infinite number of natural number divisors, thus it doesn't either.

But why is it important for 1 not to be a prime number? It's not just a matter of nitpicking. If 1 is not a prime number, then any composite number (such as 12) can be written as a product of primes in only one way (here, 2*2*3), not counting different orders. However, if 1 were a prime number, there would be infinitely many ways! We could write 12 for example, as 2*2*3, or 1*2*2*3, or 1*1*1*1*1*2*2*3. Having only one way to write a number as a product of primes is very useful when doing math.

The Unique-Prime-Factorization Theorem is so useful, that it is also called the Fundamental Theorem of Arithmetic.