Euler's Formula and Euler's Theorem

Fermat’s Theorem gives us very useful information about congruences involving exponents when the modulus is a prime number p, but one can immediately wonder about how to generalize this result to a possibly composite modulus n. In fact, the mathematician Leonard Euler developed such a generalization, called Euler’s Theorem, but the statement of the theorem required a new function which he also developed, called Euler’s Function. Euler then introduced a function which takes in a positive integer n and returns the number of elements in the range {1, 2, …, n} which are relatively prime to n.