In the Group $G= \mathbb{Z}/24\mathbb{Z}$, let $H = \langle4 + 24\mathbb{Z}\rangle$ and $N = \langle 6 + 24\mathbb{Z}\rangle$. List the elements in $H + N$ and $H \cap N$
I am aware that $G$ is the equivalence classes of the elements $\{0,1,2, \ldots, 23\}$ however I don't understand the notation of $H$ and $N$. I recognize these as cyclic groups but am not sure how to proceed from this point. Thanks in advance for any help.
The notation $H=\langle 4+24\mathbb{Z}\rangle$ means "the minimal subgroup of $G$ containing the element $4+24\mathbb{Z}$". As we are in a cyclic group, this means that $H=\{4n+24\mathbb{Z}\mid n\in\mathbb{Z}\}$. Why?
Similarly, $N=\{6m+24\mathbb{Z}\mid m\in\mathbb{Z}\}$.
Now, addition in our group is defined as: $$(a+24\mathbb{Z})+(b+24\mathbb{Z})=(a+b)+24\mathbb{Z}.$$ Hence, every element of $H+N$ has the form $4n+6m+24\mathbb{Z}$. You should then use this to describe $H+N$. Your description should be of the form $\langle p+24\mathbb{Z}\rangle$, as with the definitions of $H$ and $N$. (Hint. the key phrase is "greatest common divisor".)
To understand $H\cap N$ you need to describe all $k\in\mathbb{Z}$ such that $k=4n+24a=6m+24b$ for $a, b, m, n\in\mathbb{Z}$. Again, your description should be of the form $\langle p+24\mathbb{Z}\rangle$, and again I will give you a hint... (Hint. the key phrase is "least common multiple".)