由(x,y)=5!,可設(shè)x=(5!)a,y=(5!)b.則(a,b)=(x,y)/5!=1,即a和b互素,所以[a,b]=ab.所以[x,y]=(5!)[a,b]=(5!)ab.所以問題等價(jià)于：
求有序正整數(shù)對(duì)a,b的個(gè)數(shù),使得a和b互素,且ab=50!/5!=6*7*8*...*50.
引理：設(shè)正整數(shù)N有k個(gè)素因子,則恰有2^k個(gè)有序正整數(shù)對(duì)a,b,使得a和b互素,且ab=N.
證明：設(shè)N=(p1^n1)(p2^n2)...(pk^nk).由條件易知若pi|a,則pi^ni整除a（否則pi|a且pi|b）.所以a總等于某些pi^ni的乘積,而b等于N/a.這樣的不同取法恰好一一對(duì)應(yīng)于集合{p1^n1,p2^n2,...,pk^nk}的不同子集（空集對(duì)應(yīng)于a=1,b=N；全集對(duì)應(yīng)于a=N,b=1）,共有2^k個(gè).
6*7*8*...*50的素因子有：2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,共15個(gè).所以所求正整數(shù)對(duì)的個(gè)數(shù)=2^15=32768