Is the union all numbers divisible by 5 and divisible 3 countable? Using the method where you determine if this set has the same cardinality as the positive integers where would you start? there appears to be no straightforward equation representing the set that contains all divisors of 3 and 5, can this be done with those diagonal charts? 

insight into where to start or what to learn would be helpful
Use the Cantor Bernstien theorem:

https://en.wikipedia.org/wiki/Schr%C3%B6der%E2%80%93Bernstein_theorem

The canonical injection $X \longrightarrow \mathbb{Z}$, together with $\mathbb{Z} \longrightarrow X$ defined by $n \mapsto 5n$, where $X$ is the set of all integers divisible by 5 or 3, fulfills the requirements for the CBT.
Hint: Can you construct a bijection from the natural numbers to your set?  If you can, what does that tell you?