The Fundamental theorem of arithmethic ( also called the unique factorization theorem ) is a theorem of number theory . The theorem says that every positive integer greater than 1 can be written as a product of prime numbers ( or the integer is itself a prime number ) . In number theory , the fundamental theorem of arithmetic ( or the unique-prime-factorization theorem ) states that any integer greater than 1 can be written as a unique product ( up to ordering of the factors ) of prime numbers .