# Divisor

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A natural number ${d}$ is called a divisor of a natural number ${n}$ if there is a natural number ${k}$ such that $n=kd$ or, in other words, if $\frac nd$ 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 the main article on counting divisors. If $n=p_{1}^{\alpha_{1}} \cdot p_{2}^{\alpha_{2}}\cdot\dots\cdot p_m^{\alpha_m}$ is the prime factorization of ${n}$, then the number $d(n)$ of different divisors of $n$ is given by the formula $d(n)=(\alpha_{1} + 1)\cdot(\alpha_{2} + 1)\cdot\dots\cdot(\alpha_{m} + 1)$. It is often useful to know that this expression grows slower than any positive power of ${n}$ as $n\to\infty$. We also know that the product of the divisors of any integer $n$ is $$n^{\frac{t(n)}{2}}.$$ Another useful idea is that $d(n)$ is odd if and only if ${n}$ is a perfect square.

## Useful formulas

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