Number theory divisibility question

[Idiotinabox]

Let a, b and c be positive integers such that a^(b+c) = b^c x c Prove that b is a divisor of c, and that c is of the form d^b for some positive integer d.

I'm not sure how to solve this question at all, I need some help.

 

[chiro]

Hey Idiotinabox.

Have you come across logarithmic congruence relations? Is there a way to use with the added constraint of c MOD b = 0?

 

[Idiotinabox]

No, I have not come across them before. Could you elaborate?