Difference between revisions of "Divisor"
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− | A [[natural number]] <math> | + | A [[natural number]] <math>{d}</math> is called a '''divisor''' of a natural number <math>{n}</math> if there is a natural number <math>{k}</math> such that <math>n=kd</math> or, in other words, if <math>\frac nd</math> is also a natural number (i.e <math>d</math> divides <math>n</math>). See [[Divisibility]] for more information. |
== Notation== | == Notation== | ||
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− | See the main article on [[counting divisors]]. If <math>n= | + | See the main article on [[counting divisors]]. If <math>n=p_{1}^{\alpha_{1}} \cdot p_{2}^{\alpha_{2}}\cdot\dots\cdot p_m^{\alpha_m}</math> is the [[prime factorization]] of <math>{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(\alpha_{2} + 1)\cdot\dots\cdot(\alpha_{m} + 1)</math>. It is often useful to know that this expression grows slower than any positive power of <math>{n}</math> as <math>n\to\infty</math>. Another useful idea is that <math>d(n)</math> is [[odd integer | odd]] if and only if <math>{n}</math> is a [[perfect square]]. |
==Useful formulae== | ==Useful formulae== | ||
− | * If <math> | + | * If <math>{m}</math> and <math>{n}</math> are [[relatively prime]], then <math>d(mn)=d(m)d(n)</math> |
− | * <math> | + | * <math>{\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== | ==See also== |
Revision as of 23:43, 31 May 2008
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