Let denote a group and let . In each case show .
a. .
b. .
c. .
d. Generalize.
The order of an element of a group always divides the order of the group (so in example c., |G| can't be 12, it must be 120).
In example a., the order of g must be 1, 2, 3, 4, 6 or 12 (those being the divisors of 12. If g has order 1 then g is the identity and so g to any other power is also 1. If g has order 2 then . If g has order 3 then . So if and then the only remaining possibility is that g has order 12. Thus g generates the whole of G.
The other two examples are similar. I'll leave you to think about the generalisation.