Question

x+y/x的平方-xy+y的平方=3/7,求它的整数解

Answer 1

x^2表示x²

(x+y)/(x^2-xy+y^2)=3/7

设x+y=3t

x^2-xy+y^2=7t

t为整数

于是

x^2+2xy+y^2=9t^2

3xy=9t^2-7t

(x-y)^2=(28t-9t^2)/3

则t是3的倍数,于是,设t=3k

则(x-y)^2=28k-27k^2=k(28-27k)≥0

又k是整数,于是k=0或1

当k=0时,分母为0,舍弃,于是k=1

则x-y=±1,x+y=3t=9k=9

x=4,y=5或x=5,y=4