Prove that for any positive integer $n$, the least common multiple of the numbers $1, 2, 3, \ldots , n$ and the least common multiple of the numbers: ${n\choose 1}, {n\choose 2}, \ldots , {n\choose n}$ are equal if and only if $n + 1$ is a prime number.
My quest
$$X_n=\{1, 2, 3, \ldots , n\}, Y_n=\left\{{n\choose 1}, {n\choose 2}, \ldots , {n\choose n}\right\}.$$
After some calculus, I'm pretty sure about this assumption: $p$ prime divides one of the $X_n$ elements if and only if $p$ divides one of the $Y_n$ elements happens if and only if $n+1$ is prime. This, I think, will solve half of the problem. Anyway, I'm unable to prove or disprove the assumption. Any help is appreciated.