Difference between revisions of "Divisor"

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== Notation==
 
== Notation==
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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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>.
  
  

Revision as of 19:50, 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 $1+1=2$. 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