Another way to see this is to observe that the Fundamental Theorem of Arithmetic would be wrong (or more difficult to state succinctly) if 2 were not considered prime or if 1 were considered prime.
For example, if 2 were not prime, it would be impossible to represent the number 256 (or the number 123456) as a product of primes.
For example, if 2 were not prime, it would be impossible to represent the number 256 (or the number 123456) as a product of primes.