若x、y、z均是正整数,试说明(z^-x^-y^)^-4x^y^能被x+y+z整除.
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若x、y、z均是正整数,试说明(z^-x^-y^)^-4x^y^能被x+y+z整除.
![若x、y、z均是正整数,试说明(z^-x^-y^)^-4x^y^能被x+y+z整除.](/uploads/image/z/16604495-71-5.jpg?t=%E8%8B%A5x%E3%80%81y%E3%80%81z%E5%9D%87%E6%98%AF%E6%AD%A3%E6%95%B4%E6%95%B0%2C%E8%AF%95%E8%AF%B4%E6%98%8E%EF%BC%88z%5E-x%5E-y%5E%29%5E-4x%5Ey%5E%E8%83%BD%E8%A2%ABx%2By%2Bz%E6%95%B4%E9%99%A4.)
(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.
= (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.
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