Counting general power residues

Suppose every integer is taken to the power of a fixed integer exponent k ≥ 2 and the remainders of these powers upon division by a fixed integer n ≥ 2 are found. It is natural to ask how many distinct remainders are produced. By building on the work of Stangl, who published the k = 2 case in Mathematics Magazine in 1996, we find essentially closed formulas that allow for the computation of this number for any k. Along the way, we provide an exposition of classical results on the multiplicativity of this counting function and results on the number of remainders that are coprime to the modulus n.