  Complete residue systems play an integral role in abstract algebra and number
theory, and a description is typically found in any number theory textbook.
This note provides a concise overview of complete residue systems, including a
robust definition, several well-known results, a proof to the converse of a
well-known theorem, ancillary results pertaining to an application arising from
the study of the roots of nonnegative matrices, and extends our knowledge of
complete residue systems in relation to complete sets of roots of unity.
