An excerpt from Proof and the Art of Mathematics — the fundamental theorem of arithmetic
I like the following style of explanation of why 1 is not prime. I'll spell it out for the natural numbers. Define n to be prime if, whenever n is equal to a product of natural numbers, it is identical to one of the members of the product. Then 1 is not prime because it is equal to the empty product, but not identical to any of the members of that product, because there are none. This form of explanation is discussed further on the ncatlab page "too simple to be simple".