# Lucas' theorem

Suppose  and  are nonnegative integers and  is a prime number. Suppose  and  are the expressions of  and  in base , so that each  is in the set . if , define  for . Then, we have:

By convention,  if  or if .
In particular, we have the following: If  is a power of  and , then  is relatively prime to . For more on this special case and alternative proofs of it, see Lucas' theorem prime power case.