Answer
See explanations.
Work Step by Step
Step 1. Prove the statement is true when $n=1$: $x-y$ is a factor of $x^1-y^1$ , thus it is true for $n=1$.
Step 2. Assume the statement is true when $n=k$: we have $x-y$ is a factor of $x^k-y^k$
Step 3. Prove it is true for $n=k+1$: $x^{k+1}-y^{k+1}=xx^k-yx^k+yx^k-yy^k=x^k(x-y)+y(x^k-y^k)$. As $x-y$ is a factor of both $x^k-y^k$ and $x^k(x-y)$, it is also a factor of their sum.
Thus, the statement is also true for $n=k+1$
Step 4. Conclusion: with mathematical induction, we have proved that the statement is true for all natural numbers $n$.
