若x、y、z均是正整数,试说明(z^-x^-y^)^-4x^y^能被x+y+z整除.
问题描述:
若x、y、z均是正整数,试说明(z^-x^-y^)^-4x^y^能被x+y+z整除.
答
(z^2 - x^2 - y^2)^2 - 4x^2 * y^2
= (z^2 - x^2 - y^2 - 2xy)(z^2 - x^2 - y^2 + 2xy)
= (z^2 -(x+y)^2)(z^2 - (x-y)^2)
= (z-x-y)(z+x+y)(z-x+y)(z-x+y)
So,(z^-x^-y^)^-4x^y^能被x+y+z整除.