When Additive and Multiplicative Inverses are the Same

Summary A professor and her students were working through a routine problem on isomorphisms that required finding the multiplicative inverse of 43 modulo 50. The professor, however, simply asked for “the inverse” of 43 modulo 50 and got a surprising response. One student quickly answered that the inverse of 43 modulo 50 was 7. Thinking that the student was responding with the additive inverse, the professor restated her question and asked for the multiplicative inverse. The student checked her work and said that 7 is the multiplicative inverse of 43 mod 50. The observation that the additive and multiplicative inverse of an element could be the same led to a research project into the questions of when, why, and how frequently this phenomenon occurs. Answering these questions led to tools such asthe Chinese remainder theorem, the fundamental theorem of cyclic groups, and quadratic residues.