I am not sure I understand the wording of the first question, but if we are thinking of and as additive groups, the kernel of f is everything that gets sent to 0 by the floor function, so .

The other two questions are just giving you the additive cyclic groups respectively. Think clock arithmetic.

The first splits every natural number into two sets, the evens- and the odds .

The other splits the natural numbers into 4 distinct equivalence classes based on their remainder when dividing by 4. So by the Euclidean algorithm, you get

Hope that helps.