# Difference between revisions of "Prime factorization"

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For a positive integer <math>n</math>, the '''prime factorization''' of <math>n</math> is an expression for <math>n</math> as a product of powers of [[prime number]]s. 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. | For a positive integer <math>n</math>, the '''prime factorization''' of <math>n</math> is an expression for <math>n</math> as a product of powers of [[prime number]]s. 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 <math>{p_1}^{e_1}\cdot</math><math>{p_2}^{e_2}\cdot{p_3}^{e_3}\cdots{p_k}^{e_k} = n</math>, where ''n'' is any natural number, the <math>p_{i}</math> are prime numbers, and the <math>e_i</math> are their positive integral exponents. | ||

+ | Prime factorizations are important in many ways, for instance, to simplify [[fraction]]s. | ||

+ | ===Example Problem=== | ||

The prime factorization of 378 is <math>2^1\cdot3^3\cdot7^1</math>. | The prime factorization of 378 is <math>2^1\cdot3^3\cdot7^1</math>. | ||

− | + | ===See also=== | |

− | + | ||

− | + | *[[Divisor]] |

## Revision as of 17:38, 24 June 2006

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 *n* is any natural number, the are prime numbers, and the are their positive integral exponents.
Prime factorizations are important in many ways, for instance, to simplify fractions.

### Example Problem

The prime factorization of 378 is .