I'm not going to list the choices here, mainly because I just want the general idea. If I generalize the question and was given $n$ different integers divide some integer $r$, how do I determine what else it is divisible by?
My initial attempt at the problem was that it should be divisible by $2$ (since $8$ is divisible by $2$), and $3$ and $5$ (since $15$ is divisible by those). Then I thought the number should be divisible by $2, 3,$ and $5$. But apparently that's incorrect.
In general, if an integer is divisible by $a_1,a_2,\ldots,a_k$, then the integer is divisible by $\text{lcm}(a_1,a_2,\ldots,a_k)$. The proof of this claim immediately follows from the fact that if $a \vert bc$ and $\gcd(a,b) = 1$, then $a \vert c$, which follows immediately from the definition of $\gcd$.
Hence, in your case, the integer is divisible by $120$.
You are on the right track, but being divisible by $8$ implies more than being divisible by $2$.
If $a$ is divisible by $b$, then $a$ is also divisible by all the factors of $b$. If $a$ is divisible by $b$ and $c$, then $a$ is divisible by products of the factors of $b$ and $c$. Does that help?
You could also calculate the smallest number that is divisible by both $8$ and $15$ and find what numbers divide that number.
