# 因數

（重定向自因子

## 定义

${\displaystyle a,b}$  满足 ${\displaystyle a\in \mathbb {N} ^{*},b\in \mathbb {N} }$ . 若存在 ${\displaystyle q\in \mathbb {N} }$  使得 ${\displaystyle b=aq}$ , 那么就说 ${\displaystyle b}$ ${\displaystyle a}$ 倍数${\displaystyle a}$ ${\displaystyle b}$ 约数。这种关系记作 ${\displaystyle a|b}$ ,读作“${\displaystyle a}$  整除 ${\displaystyle b}$ ”.

## 性质

• ${\displaystyle a|b,\;b|c}$  那么 ${\displaystyle a|c}$ .
• ${\displaystyle a|b,\;a|c}$ ${\displaystyle x,y\in \mathbb {Z} }$ , 有 ${\displaystyle a|(bx+cy)}$ .
• ${\displaystyle a|b}$ , 设 ${\displaystyle t\not =0}$ , 那么 ${\displaystyle (ta)|(tb)}$ .
• ${\displaystyle b=qd+c}$ , 那么 ${\displaystyle d|b}$ 充要条件${\displaystyle d|c}$
• ${\displaystyle x,y\in \mathbb {Z} }$  满足 ${\displaystyle ax+by=1,\;a|n.\;b|n}$  那么 ${\displaystyle ab|n}$ .

${\displaystyle \because a|n,\;b|n\quad \therefore ab|bn,\;ab|an\quad \therefore ab|(anx+bny)}$

${\displaystyle \because ax+by=1\quad \therefore ab|n}$

## 相关定理

### 整数的唯一分解定理

${\displaystyle A=\prod _{i=1}^{n}p_{i}^{a_{i}}}$ , 其中 ${\displaystyle p_{i}}$  是一个素数.

### 因数个数

${\displaystyle N}$  唯一分解为 ${\displaystyle N=p_{1}^{a_{1}}\times p_{2}^{a_{2}}\times p_{3}^{a_{3}}\times \cdots \times p_{n}^{a_{n}}=\prod _{i=1}^{n}p_{i}^{k_{i}}}$ , 则 ${\displaystyle d(N)=(a_{1}+1)\times (a_{2}+1)\times (a_{3}+1)\times \cdots \times (a_{n}+1)=\prod _{i=1}^{n}\left(a_{i}+1\right)}$ .

### 因数和

${\displaystyle N}$  唯一分解为 ${\displaystyle N=p_{1}^{a_{1}}\times p_{2}^{a_{2}}\times p_{3}^{a_{3}}\times \cdots \times p_{n}^{a_{n}}=\prod _{i=1}^{n}p_{i}^{k_{i}}}$ , 则 ${\displaystyle \sigma (N)=\prod _{i=1}^{n}\left(\sum _{j=0}^{a_{i}}p_{i}^{j}\right)}$ .

{\displaystyle {\begin{aligned}\sigma (N)&={\frac {p_{1}^{a_{1}+1}-1}{p_{1}-1}}\times {\frac {p_{2}^{a_{2}+1}-1}{p_{2}-1}}\times \cdots \times {\frac {p_{n}^{a_{n}+1}-1}{p_{n}-1}}&\end{aligned}}}

{\displaystyle {\begin{aligned}\sigma (2646)&=(1+2)\times (1+3+9+27)\times (1+7+49)\\&={\frac {2^{2}-1}{2-1}}\times {\frac {3^{4}-1}{3-1}}\times {\frac {7^{3}-1}{7-1}}\\&=3\times 40\times 57\\&=6840\end{aligned}}}

## 其他

• 1是所有整數的正因數，-1是所有整數的負因數，因為${\displaystyle x=1x=-1\times (-x)}$

• 質數${\displaystyle p}$ 只有2個正因數：1, ${\displaystyle p}$ ${\displaystyle p}$ 平方數只有三個正因數：1, ${\displaystyle p}$ , ${\displaystyle p^{2}}$