Prove for every natural number n, x-y divides (x^n) - (y^n)

Prove for every natural number n, x-y divides (x^n) - (y^n)

This is another proof by induction, so first we prove the first case, which for the natural numbers is 1.

x - y divides (x^1) - (y^1) = 1(x-y) so this is true.

Now by induction we must show that x-y divides x^(n+1) - y^(n+1)

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

= x(x^n) - y(y^n)

= x(x^n) - y(x^n) + y(x^n) - y(y^n)

= (x-y)(x^n) + y(x^n - y^n)

so now we have a form where x-y divides the first term (since x-y is a factor) and divides the second term((x^n - y^n)) by the hypothesis of induction.

Thus x-y divides (x^n) - (y^n) for all natural numbers n.