Let $H = \mathbb Z/2\mathbb Z$ and $K = \mathbb Z/4\mathbb Z$
Find the elements of the subgroup $$H \cap K$$
My approach:
Since $H\cap K$ is sugroup of H and K, it contains the trivial element, the identity element. Since 2 and 4 are not co-prime, there could exist other element in the intersection.
I'm not getting how to find the rest of the elements. Any help ?
The question makes no sense, since as you present them $H$ and $K$ are abstract groups and there is no intersection between such.
If you want to speak of an intersection, you need to embed them into one larger group, (to see them as subgroups of a larger group).
There you will be able to discuss the intersection. It is also possible to discuss what the intersection can possibly be, given the orders of the elements.