# 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 | + | The form of a prime factorization is |

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+ | <math>\displaystyle n = {p_1}^{e_1} \cdot {p_2}^{e_2}\cdot{p_3}^{e_3}\cdots{p_k}^{e_k}</math> | ||

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+ | where <math>\displaystyle n</math> is any natural number, the <math>p_{i}</math> are prime numbers, and the <math>e_i</math> are their positive integral exponents. | ||

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Prime factorizations are important in many ways. One instance is to simplify [[fraction]]s. | Prime factorizations are important in many ways. One instance is to simplify [[fraction]]s. | ||

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

## Revision as of 15:14, 29 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 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.

### Example Problem

The prime factorization of 378 is .

## Resources

### Books

### Games