From: Dan Asimov

But I wanted to chime in that 24 is my choice for most interesting integer, too.

Partly because, among all the multiplicative groups of rings

     G_n  :=  (Z/nZ)*,

24 is the largest n for which k | n implies that all elements of G_k are squares.

In G_3, only 1 is a square, and 2 isn't. In fact, for any k | 24, G_k *only* has 1 as a square. Phil