Divisibilty of a functional equation

Question

I found this question in a mathematical problems book:

Let $f(x)$ is a polynomial such that $f(x^n)$ is divisible by $x-1$. Prove that $f(x^n)$ is divisible by $x^n-1.$

Can anybody help me?

Answer 1

$$(x-1) \mid f(x^n) \implies f(1)=0 \implies (x-1) \mid f(x) \implies (x^n-1) \mid f(x^n) $$