设a,b为整数.既能整除a,又能整除b的正整数称为a,b的公因数,其最大者称为a,b的最大公因数,记作\( \left( {a,b} \right) \) ,即\[ \left( {a,b} \right) = \max \{ \begin{array}{*{20}c} {c|} & {c|a,c|b} \\ \end{array}\} \] 特别当\( \left( {a,b} \right) = 1 \) ,称a,b互素.
设a,b为正整数.a,b都能整除的正整数称为a,b的公倍数,其最小者称为a,b的最小公倍数,记作\( \left[ {a,b} \right] \),即\( \left[ {a,b} \right] = \min \{ \begin{array}{*{20}c} {\alpha |} & {a|\alpha ,b|\alpha } \\ \end{array}\} \)
设为n个正整数,用归纳法定义其最大公因数为 \[ \left( {a_1 ,a_2 , \cdots ,a_n } \right) = \left( {\left( {a_1 , \cdots ,a_{n - 1} } \right)a_n } \right) \] 其最小公倍数为\[ \left[ {a_1 ,a_2 , \cdots ,a_n } \right] = \left[ {\left[ {a_1 , \cdots ,a_{n - 1} } \right],a_n } \right] \]
最大公因数与最小公倍数具有下列性质:
1°存在整数x,y,使得(a,b)=ax+by.并可由辗转相除法具体求出x,y.
2°对任意二整数x,y,必有\( \left( {a,b} \right)|ax + by \) .
3°若e|a,e|b,则e|(a,b).
4°若c>0,(a,b)=d,则(ac,bc)=dc.若c>0,c|d,则\( \left( {\frac{a}{c},\frac{b}{c}} \right) = \frac{d}{c} \)
5°若a,b为二正整数,为它们的素因数,且标准分解式分别为 \[ \begin{array}{*{20}c} {a = p_1 ^{a_1 } \cdots p_s ^{a_s } ,a_i \ge 0} & {\left( {1 \le i \le s} \right)} \\ \end{array} \] \[ \begin{array}{*{20}c} {b = p_1 ^{b_1 } \cdots p_s ^{b_s } ,b_i \ge 0} & {\left( {1 \le i \le s} \right)} \\ \end{array} \] \[ p_1 < \cdots p_s \] 则 \[ \begin{array}{*{20}c} {\left( {a,b} \right) = p_1 ^{c_1 } \cdots p_s ^{c_s } } & {c_i = \min \left( {a_i ,b_i } \right)} & {\left( {1 \le i \le s} \right)} \\ \end{array} \] \[ \begin{array}{*{20}c} {\left[ {a,b} \right] = p_1 ^{c_1 } \cdots p_s ^{c_s } } & {c_i = \max \left( {a_i ,b_i } \right)} & {\left( {1 \le i \le s} \right)} \\ \end{array} \] 6°\( \left[ {a,b} \right]\left( {a,b} \right) = ab \)
7°若为互素的正整数,即\( \left( {a_1 ,a_2 , \cdots ,a_n } \right) = 1 \),则\[ \left( {a_1 ,a_2 } \right) \cdots \left( {a_1 ,a_n } \right) \le a_1 ^{n - 2} \]