Difference between revisions of "Divisor"
(→Notation) |
(→Notation) |
||
Line 6: | Line 6: | ||
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>. | 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>. | ||
− | We also know that the product of the divisors of any integer <math>n</math> is <cmath>n^{\frac{t(n)}{2}}</cmath> | + | We also know that the product of the divisors of any integer <math>n</math> is <cmath>n^{\frac{t(n)}{2}}.</cmath> |
Another useful idea is that <math>d(n)</math> is [[odd integer | odd]] if and only if <math>{n}</math> is a [[perfect square]]. | Another useful idea is that <math>d(n)</math> is [[odd integer | odd]] if and only if <math>{n}</math> is a [[perfect square]]. | ||
Revision as of 14:01, 27 April 2021
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
.
We also know that the product of the divisors of any integer
is
Another useful idea is that
is odd if and only if
is a perfect square.
Useful formulas
- If
and
are relatively prime, then