Does lcm of multiple numbers also divide any common multiple of these numbers?

Question

I know that this is true for two numbers, but does this also hold for more than two? I.e. if $m$ is a common multiple of several numbers $n_1, \ldots , n_k$, does it hold that lcm$(n_1,\ldots,n_k)$ divides $m$? If so, how would one go about proving that? Maybe considering pairs of numbers of the set?

Answer 1

Hint Consider the prime factor decomposition and note that the exponent $p(x)$ of a prime $p$ dividing the lcm $x$ of numbers $x_1,...,x_k$ is given by $\max \{p(x_1), ... , p(x_k)\}$. What can you say about $p(y)$ for a common multiple $y$ of $x_1,...,x_k$?

Answer 2

By definition, $\DeclareMathOperator{\lcm}{lcm}\lcm(n_1,\dots,n_k)$ is the positive generator of the (principal) ideal of common multiples of $n_1,\dots,n_k$: $$n_1\mathbf Z\cap\dots\cap n_k\mathbf Z=\lcm(n_1,\dots,n_k)\mathbf Z,$$ so the answer is obvious.