How to find the generators of a principal Ideal?

Hands-on approach: We know that the order of $ain G$ an element equals the order of the ideal it generates. In this case, $|I| = 8$, so $I = (a) iff ncdot abmod 24 neq 0$ for $n = 1,dots,7$.

Since elements of $I$ are of the form $mcdot 3$, this is looking for $m = 1,dots,7$ such that $ncdot m bmod 8 neq 0$ for $n = 1,dots,7$:

$$ncdot m bmod 8 = 0 iff (n,m) in {(4,2),(2,4),(4,4),(6,4),(4,6)}$$

So $6,12,18$ don't generate $I$, and $3,9,15,21$ generate it.