Divisor

Definition

Any natural number $\displaystyle{d}$ is called a divisor of a natural number $\displaystyle{n}$ if there is a natural number $\displaystyle{k}$ such that $n=kd$ or, in other words, if $\displaystyle\frac nd$ is also a natural number (i.e $d$ divides $n$ ). See Divisibility for more information.

Notation

A common notation to indicate a number is a divisor of another is $n|k$ . This means that $n$ divides $k$ .

See main article, Counting divisors. If $n=p_1^{\alpha_1}\cdot\dots\cdot p_n^{\alpha_n}$ is the prime factorization of $\displaystyle{n}$ , then the number $d(n)$ of different divisors of $n$ is given by the formula $d(n)=(\alpha_1+1)\cdot\dots\cdot(\alpha_n+1)$ . It is often useful to know that this expression grows slower than any positive power of $\displaystyle{n}$ as $\displaystyle n\to\infty$ . Another useful idea is that $d(n)$ is odd if and only if $\displaystyle{n}$ is a perfect square.

Useful formulae

If $\displaystyle{m}$ and $\displaystyle{n}$ are relatively prime, then $d(mn)=d(m)d(n)$
$\displaystyle{\sum_{n=1}^N d(n)=\left\lfloor\frac N1\right\rfloor+\left\lfloor\frac N2\right\rfloor+\dots+\left\lfloor\frac NN\right\rfloor= N\ln N+O(N)}$

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Divisor

Contents

Definition

Notation

Useful formulae

See also