Divisor: Difference between revisions
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==Definition== | |||
Any [[natural number]] <math>\displaystyle{d}</math> is called a divisor of a natural number <math>\displaystyle{n}</math> if there is a natural number <math>\displaystyle{k}</math> such that <math>n=kd</math> or, in other words, if <math>\displaystyle\frac nd</math> is also a natural number. See [[Divisibility]] for more information. | Any [[natural number]] <math>\displaystyle{d}</math> is called a divisor of a natural number <math>\displaystyle{n}</math> if there is a natural number <math>\displaystyle{k}</math> such that <math>n=kd</math> or, in other words, if <math>\displaystyle\frac nd</math> is also a natural number (i.e <math>d</math> divides <math>n</math>). See [[Divisibility]] for more information. | ||
== Notation== | |||
A common notation to indicate a number is a divisor of another is <math>n|k</math>. This means that <math>n</math> divides <math>k</math>. | |||
See main article, [[Counting divisors]]. If <math>n=p_1^{\alpha_1}\cdot\dots\cdot p_n^{\alpha_n}</math> is the [[prime factorization]] of <math>\displaystyle{n}</math>, then the number <math>d(n)</math> of different divisors of <math>n</math> is given by the formula <math>d(n)=(\alpha_1+1)\cdot\dots\cdot(\alpha_n+1)</math>. It is often useful to know that this expression grows slower than any positive power of <math>\displaystyle{n}</math> as <math>\displaystyle n\to\infty</math>. Another useful idea is that <math>d(n)</math> is odd if and only if <math>\displaystyle{n}</math> is a perfect square. | See main article, [[Counting divisors]]. If <math>n=p_1^{\alpha_1}\cdot\dots\cdot p_n^{\alpha_n}</math> is the [[prime factorization]] of <math>\displaystyle{n}</math>, then the number <math>d(n)</math> of different divisors of <math>n</math> is given by the formula <math>d(n)=(\alpha_1+1)\cdot\dots\cdot(\alpha_n+1)</math>. It is often useful to know that this expression grows slower than any positive power of <math>\displaystyle{n}</math> as <math>\displaystyle n\to\infty</math>. Another useful idea is that <math>d(n)</math> is odd if and only if <math>\displaystyle{n}</math> is a perfect square. | ||
==Useful formulae== | |||
* If <math>\displaystyle{m}</math> and <math>\displaystyle{n}</math> are [[relatively prime]], then <math>d(mn)=d(m)d(n)</math> | * If <math>\displaystyle{m}</math> and <math>\displaystyle{n}</math> are [[relatively prime]], then <math>d(mn)=d(m)d(n)</math> | ||
* <math>\displaystyle{\sum_{n=1}^N d(n)=\left\lfloor\frac N1\right\rfloor+\left\lfloor\frac N2\right\rfloor+\dots+\left\lfloor\frac NN\right\rfloor= N\ln N+O(N)}</math> | * <math>\displaystyle{\sum_{n=1}^N d(n)=\left\lfloor\frac N1\right\rfloor+\left\lfloor\frac N2\right\rfloor+\dots+\left\lfloor\frac NN\right\rfloor= N\ln N+O(N)}</math> | ||
==See also== | |||
*[[ | * [[Divisor function]] | ||
*[[Number theory]] | * [[Number theory]] | ||
*[[GCD]] | * [[GCD]] | ||
*[[Divisibility]] | * [[Divisibility]] | ||
Revision as of 22:19, 28 July 2006
Definition
Any natural number 
is called a divisor of a natural number 
if there is a natural number 
such that 
or, in other words, if 
is also a natural number (i.e 
divides 
). See Divisibility for more information.
Notation
A common notation to indicate a number is a divisor of another is 
. This means that divides 
.
See main article, Counting divisors. If 
is the prime factorization of , then the number 
of different divisors of is given by the formula 
. It is often useful to know that this expression grows slower than any positive power of as 
. Another useful idea is that is odd if and only if is a perfect square.
Useful formulae
- If
and 
are relatively prime, then 
and 
