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 
.
