Question

x^n - y^n 可以被 x-y 整除.

Answer 1

n=1,x-y能被x-y整除成立

假设n=k时,k>=1

x^k-y^k能被x-y整除成立

则n=k+1

x^(k+1)-y^(k+1)

=x^(k+1)-y^(k+1)-x*y^k+x^k*y+x*y^k-x^k*y

=[x^(k+1)-x^k*y]-[y^(k+1)-x*y^n]+x*y^k-x^k*y

=[x^(k+1)-x^k*y]-[y^(k+1)-x*y^n]+x*y^k-x^k*y

=x^k(x-y)-y^k(x-y)+x*y^k-x^k*y

=(x-y)(x^k-y^k)-x^(k-1)*y^(k-1)*(x-y)

=(x-y)[(x^k-y^k)-x^(k-1)*y^(k-1)]

能被xy整除

综上

x^n-y^n 可以被 x-y 整除.