Herein, we offer a gentle yet rigorous introduction to both modular arithmetic and the elementary combinatorics of the common shift mapping, culminating in self-contained and accessible proofs of various cases of a general Euler Totient Function Theorem, Fermat's Little Theorem included. The reader may consider this an homage to T.P.

Kirkman and W.S.B. Woolhouse for their pioneering work in elementary combinatorics [2], [3].

We recommend [1] both as a primer on the shift mapping and as an engaging example of elementary mathematics transmogrified