Is it possible to know if sums of powers of a number is divisible by another number?

Question

Is there a way to find whether a number (say $A$) formed by summing powers of another number (say $B$) is divisible by another number $C$? $A$ is a number like, for example, $B^1+B^3$. We can use a power of $B$ at most one time.

I don't want to know the number $A$. I just want to know that is $A$ is divisible by $C$.

Any help is appreciated.

edit: i found out that the answer is always yes :D

Answer 1

If B and C are coprime, $B^{\phi(c)}=1 \pmod C$, where $\phi(n)$ is Euler's totient function. In this case there is always such an A, take $\sum_{i=1}^CB^{i\phi(C)}$. If they are not coprime I believe you can do this with the $B/gcd(B,C)$