Isn't "every positive integer can be written as a product of primes greater than 1 in one and only one way" incorrect? A prime number is a only product of itself * 1, isn't it?
Mathematicians generally feel that a single number qualifies as a "product of 1 number." So 7 can be written as just 7 which is still considered a product of prime(s). This is purely a convention thing to make it so theorems can be stated more succinctly, as with not counting 1 as prime.
