Difference between revisions of "Divisor"

Revision as of 11:06, 8 July 2016

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 can also count the product of divisors of any integer $n$ with this formula: $n^{\frac{t(n)}{2}}$ 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 $d(mn)=d(m)d(n)$
${\sum_{n=1}^N d(n)=\left\lfloor\frac N1\right\rfloor+\left\lfloor\frac N2\right\rfloor+\dots+\left\lfloor\frac NN\right\rfloor= N\ln N+O(N)}$

@@ Line 5: / Line 5: @@
-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]].
+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 can also count the product of divisors of any integer <math>n</math> with this formula: <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]].
 ==Useful formulas==

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Difference between revisions of "Divisor"

Revision as of 11:06, 8 July 2016

Notation

Useful formulas

See also