Fundamental theorem of arithmetic

In number theory, the fundamental theorem of arithmetic (or unique factorization theorem) states that every natural number greater than 1 can be written as a unique product of prime numbers. For instance,

There are no other possible factorizations of 6936 or 1200 into prime numbers. The above representation collapses repeated prime factors into powers for easier identification. Because multiplication is commutative, the order of factors is irrelevant and usually written from smallest to largest.

Many authors take the natural numbers to begin with 0, which has no prime factorization. Thus Theorem 1 of Hardy & Wright (1979) takes the form, “Every positive integer, except 1, is a product of primes”, and Theorem 2 (their "Fundamental") asserts uniqueness. The number 1 is not itself prime, but since it is the product of no numbers, it is often convenient to include it in the theorem by the empty product rule. (See, for example, Calculating the GCD.)

Hardy & Wright define an abnormal number to be a hypothetical number that does not have a unique prime factorization. They prove the fundamental theorem of arithmetic by proving that there does not exist an abnormal number.