Cyclic Groups w/ 2 generators

[linzedun]

The question goes like this: Determine all cyclic groups that have exactly two generators?

 

I know that the answer is, Z, Z3, Z4, and Z6... however, I don't understand where that comes from, how do you know those are the only ones? Why isn't U(10) one? My test is tomorrow, I would really appreciate your help!!!

[matt grime]

i think there's a mistake here does the question really ask exactly that? A cyclic group only ever has one generator, that is the definition of cyclic.