Unique factorization

In mathematics, the unique factorization theorem, also known as the fundamental theorem of arithmetic states that every positive whole number can be expressed as a product of prime numbers in essentially only one way. It reveals that one may consider the prime numbers as "atoms" from which all other whole numbers can be assembled through multiplication. Unique factorization is the foundation for most of the structure of whole numbers as described by elementary number theory. The formulation of many results would either be nonsensical, or at least more complicated, if unique factorization did not hold. For instance, the assumption that many electronic financial transactions are secure is based on the belief that the product of two very large prime numbers is difficult to factor. If several other prime factorizations were possible, perhaps some having small prime factors, finding a factorization might be much easier and such security methods would be ineffective.