Given that $a,b,n$ are positive integers. Assume that for any positive integer $k\neq b, (k-b)\mid(k^n-a)$, the which of the following must be true?

Question

(From Math Challenge II Number Theory packet)

Given that $a,b,n$ are positive integers. Assume that for any positive integer $k\neq b, (k-b)\mid(k^n-a)$, the which of the following must be true?

(A) $a>b^n$

(B) $a<b^n$

(C) $a=b^n$

(D) It depends.

I'm not sure how to approach this problem. Can someone please explain?

Answer 1

Since $k - b$ divides $k^n-b^n$ for any $k$, it also divides $$ (k^n - a) - (k^n - b^n) = b^n - a $$ for any $k \neq b$. Therefore $b^n - a$ has infinitely many distinct divisors, and so $b^n - a = 0$, which is (C).