Let $A$ be a set of all complex numbers $z$ such that $z^24=1$ and let $B$ be the set of all complex numbers $w$ such that $w^54=1.$ That is: \begin{align*}A&=\{z\;|\;z^{24}=1\}\\ B&=\{z\;|\;z^{54}=1\}\\ \end{align*} Finally, let $C$ be the set of all complex numbers that can be formed by multiplying an element of $A$ by an element of $B$:
$C=\{zw\;|\;z\in A,w\in B\}.$
How many distinct elements are there in $C$?
I'm pretty stuck on this problem and don't know where to go. Any help would be very much appreciated!
This one gets easier after looking at it in its abstract content.
You have a finite abelian group $G$ (the one Jack Yoon mentioned in his comments) and two subgroups $A,B \subset G$. The question is: How to determine |AB|?
The answer is the exact sequence
$$1 \to A \cap B \to A \times B \to AB \to 1$$
with the first map given by $x \mapsto (x,x^{-1})$ (the second map should be more obvious).
This shows $|AB| = \frac{|A||B|}{|A \cap B|}$. Note that $|A \cap B|$ is often easier to determine than $|AB|$, especially in your case.