Question

【題目】分母是385的最簡(jiǎn)真分?jǐn)?shù)有____個(gè);它們的和是____.

Answer 1

【答案】240 120

【解析】

因385=5×7×11,在1至385中,5的倍數(shù)有(個(gè));7的倍數(shù)有

=55(個(gè));11的倍數(shù)有(個(gè)).35(5×7)的倍數(shù)有(個(gè));55(5×11)的倍數(shù)有(個(gè));77(7×11)的倍數(shù)有(個(gè));385的倍數(shù)有一個(gè).

由容斥原理知,是5或7或11的倍數(shù)的數(shù)的個(gè)數(shù)是77+55+35-11-7-5+1=145

(個(gè)).故與5,7,11都互質(zhì)的數(shù)有385-145=240(個(gè)),即以385為分母的真分?jǐn)?shù)中,最簡(jiǎn)分?jǐn)?shù)有240個(gè).

因當(dāng)是最簡(jiǎn)分?jǐn)?shù)時(shí),也是最簡(jiǎn)分?jǐn)?shù)且其和為1,即最簡(jiǎn)真分?jǐn)?shù)是成對(duì)出現(xiàn)的,且每對(duì)兩數(shù)之和為1.從而240個(gè)最簡(jiǎn)真分?jǐn)?shù)可分成120對(duì),其和為120.