Gapplet

For given m > 1, we shall give the structure of the multiplicative group (Z/mZ)^* of invertible elements of Z/mZ.

When you try different m, you may see some patterns.

If m = pq, with gcd(p,q) = 1, then, according to the Chinese Remainder Theorem, Z/mZ is isomorphic to Z/pZ×Z/qZ. Then, by the present theorem, (Z/mZ)^* is isomorphic to (Z/pZ)^* × (Z/qZ)^*.

Also, by Fermat's little theorem, (Z/pZ)^* = C_p-1 if p is a prime.

This gives some insight into the structure of the multiplicative group (Z/mZ)^*, but not the full picture. For instance, it tells us that (Z/10Z)^* = C₄, but does not give a clue about (Z/8Z)^*.