已知正整数a、b、c、d满足等式a/c=b/d=ab+1/cd+1.证明a=c,b=d

问题描述:

已知正整数a、b、c、d满足等式a/c=b/d=ab+1/cd+1.证明a=c,b=d

设a/c=b/d=﹙ab+1﹚/﹙cd+1﹚=k,则:①a/c=k,∴a=ck,②b/d=k,∴b=dk,③﹙ab+1﹚/﹙cd+1﹚=k,将①②代入③化简得:﹙k-1﹚﹙cdk-1﹚=0,分两种情况讨论:⑴k-1=0,则k=1,∴a=c,b=d.⑵cdk-1=0,则k=1/cd,...