supposedly first meant to show that a^m is an element of H for

any natural number m then use Lagrange's Theorem to deduce that the order of any

element must divide the order of the group.

Well... First of all, the subgroup generated by is .

You can claim that , and set out to prove this as follows:

Since , for a random integer

use Euclidean division to obtain for some integers with . Then .

So, all the elements of

are contained in its subset and subgroup, . Therefore, .