Difference between revisions of "Prime factorization"

(added resources)
m (organized a little)
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.
+
The form of a prime factorization is  
 +
 
 +
 
 +
<math>\displaystyle n = {p_1}^{e_1} \cdot {p_2}^{e_2}\cdot{p_3}^{e_3}\cdots{p_k}^{e_k}</math>  
 +
 
 +
 
 +
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.
 +
 
 
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.
 +
 
===Example Problem===
 
===Example Problem===
  

Revision as of 15:14, 29 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


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


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.

Example Problem

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


Resources

Books

Games


See also

Invalid username
Login to AoPS