# Difference between revisions of "Divisor"

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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]]. | ||

## Latest 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