Difference between revisions of "Divisor"
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− | 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 integer | odd]] if and only if <math>\displaystyle{n}</math> is a perfect square. |
==Useful formulae== | ==Useful formulae== |
Revision as of 15:29, 12 October 2006
A 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