# Prime factorization

For a positive integer $n$, the prime factorization of $n$ is an expression for $n$ 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

$\displaystyle n = {p_1}^{e_1} \cdot {p_2}^{e_2}\cdot{p_3}^{e_3}\cdots{p_k}^{e_k}$

where $\displaystyle n$ is any natural number, the $p_{i}$ are prime numbers, and the $e_i$ are their positive integral exponents.

Prime factorizations are important in many ways. One instance is to simplify fractions.

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### Example Problem

The prime factorization of 378 is $2^1\cdot3^3\cdot7^1$.