Difference between revisions of "Divisor"

Latest revision as of 18:50, 29 August 2023

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 $d(mn)=d(m)d(n)$
${\sum_{n=1}^N d(n)=\left\lfloor\frac N1\right\rfloor+\left\lfloor\frac N2\right\rfloor+\dots+\left\lfloor\frac NN\right\rfloor= N\ln N+O(N)}$

Revision as of 18:50, 29 August 2023 (view source) Nevergonnagiveup (talk \| contribs) (→‎Notation) ← Older edit		Latest revision as of 18:50, 29 August 2023 (view source) Nevergonnagiveup (talk \| contribs) (→‎Notation)
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	== Notation==		== Notation==
−	A common notation to indicate a number is a divisor of another is <math>~~1+1=2~~</math>. This means that <math>n</math> divides <math>k</math>.	+	A common notation to indicate a number is a divisor of another is <math>n\|k</math>. This means that <math>n</math> divides <math>k</math>.

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Difference between revisions of "Divisor"

Latest revision as of 18:50, 29 August 2023

Notation

Useful formulas