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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.

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You are on the right track, but being divisible by $8$ implies more than being divisible by $2$.

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    Ah! I figured it out just now. Thank you2012-12-20
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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 $120$.

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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.

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    Duly noted. I tend to break things down and over-simplify since I'm used to tutoring High School students. Based on the complexity level of the original question and the presence of the homework tag, I felt that it was reasonable to present$a$non-rigorous but more understandable answer.2012-12-20