The Converse of Fermat's Theorem

We know that if n is prime and a prime to n, then

a^n-1 = 1(mod. n).

This is known as Fermat's Theorem                               ... (A)

The question is; Does this theorem hold good when n is composite and if so, for what values of n ? It is proposed in this note to indicate methods of solving the congruence a^n-1 = l (mod. n), where n is composite and a prime to n.