For a positive integer , the prime factorization of is an expression for as a product of powers of prime numbers. An important theorem of number theory called the Fundamental Theorem of Arithmetic tells us that every positive integer has a unique prime factorization, up to changing the order of the terms. The form of a prime factorization is
where is any natural number, the are prime numbers, and the are their positive integral exponents.
Prime factorizations are important in many ways. One instance is to simplify fractions.
The prime factorization of 378 is .