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约数

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是所有整数的正约数，-1是所有整数的负约数，因为${\displaystyle x=1x=-1\times (-x)}$

• 素数${\displaystyle p}$只有2个正约数：1, ${\displaystyle p}$${\displaystyle p}$平方数只有三个正约数：1, ${\displaystyle p}$, ${\displaystyle p^{2))$

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