Why was 1 considered as prime years ago?

Question

I've seen on Maths Is Fun that years ago, 1 was considered as prime, but now, it is not. How did this happen? I know that a prime number has only two factors, 1 and itself, and we have 1, which is also itself. Is this why? Tell me what you think. I also know that this would make the prime factorization too continuous: 1 x 1 x 1 x 1 x 1... x 3 x 3=9 and that would not be good.

Answer 1

The Fundamental Theorem of Arithmetic states (in paraphrase) that all integers greater than $1$ have only one prime factorization, and that factorization is unique to that number. So $8$ has only one prime factorization, $2^3$, and no other positive integer has the prime factorization $2^3$. If we say $1$ is prime our statement falls apart. $8$ now has an infinite number of prime factorizations: $2^3$, $1*2^3$, $1^2*2^3$, etc... Thus $1$ is not prime because it violates the Fundamental Theorem of Arithmetic.