Let T(m) denote the set of the numbers less than and relatively prime to m, where is the totient function. Define

(1)

Then a theorem of Lagrange states that

(2)

for p an odd prime (Hardy and Wright 1979, p. 98).

This can be generalized as follows. Let p be an odd prime divisor of m and the highest power which divides m, then

(3)

and, in particular,

(4)

Now, if m > 2 is even and is the highest power of 2 that divides m, then

(5)

and, in particular,

(6)

Congruence, Leudesdorf Theorem

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