Why is it important that the number one is not a prime number?

Answer 1

In number theory, the fundamental theorem of arithmetic, also called the unique factorization theorem or the unique-prime-factorization theorem, states that every integergreater than 1[note 1] either is prime itself or is the product of prime numbers, and that this product is unique, up to the order of the factors.[3][4][5] For example,

1200 = 24 × 31 × 52 = 3 × 2 × 2 × 2 × 2 × 5 × 5 = 5 × 2 × 3 × 2 × 5 × 2 × 2 = etc.

The theorem is stating two things: first, that 1200 canbe represented as a product of primes, and second, no matter how this is done, there will always be four 2s, one 3, two 5s, and no other primes in the product.

The requirement that the factors be prime is necessary: factorizations containing composite numbers may not be unique (e.g. 12 = 2 × 6 = 3 × 4).

This theorem is one of the main reasons for which 1 is not considered as a Prime number: if 1 were prime, the factorization would not be unique, as, for example, 2 = 2×1 = 2×1×1 = ...