## Bauer's Identical Congruence

Let denote the set of the numbers less than and Relatively Prime to , where is the Totient Function. Define

 (1)

A theorem of Lagrange states that
 (2)

This can be generalized as follows. Let be an Odd Prime Divisor of and the highest Power which divides , then
 (3)

and, in particular,
 (4)

Furthermore, if is Even and is the highest Power of 2 that divides , then
 (5)

and, in particular,
 (6)

Hardy, G. H. and Wright, E. M. Bauer's Identical Congruence.'' §8.5 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 98-100, 1979.