so, now see what happens when somebody says that he/she knows a positive integer, greater than 1, which can "not" be written as a product of primes. in that case we ask him/her to mention "all" the numbers, greater than 1, that can "not" be written as a product of primes. one of these numbers must be the "smallest": let's call it "n". of course, this number "n" cannot be 1. further, it cannot be a prime number, because a prime number is a 'product' of a single prime: itself. so it "must" be a product of numbers. thus-