Difference between revisions of "Prime factorization"

m (proof reading)
Line 1: Line 1:
 
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>.
  
<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.
+
===See also===
Prime factorizations are important in many ways, for instance, to simplify [[fractions]].
+
 
===Example Problem===
+
*[[Divisor]]

Revision as of 17:38, 24 June 2006

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 ${p_1}^{e_1}\cdot$${p_2}^{e_2}\cdot{p_3}^{e_3}\cdots{p_k}^{e_k} = n$, where 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, for instance, to simplify fractions.

Example Problem

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

See also

Invalid username
Login to AoPS