Difference between revisions of "Divisor"
m (→Notation) |
m |
||
| Line 5: | Line 5: | ||
| − | See main article | + | See the main article on [[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 18:30, 20 November 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 the main article on 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 
