Property of the least common multiple

Question

Let $m$ and $n$ two positive integers with $gcd(m,n)=1$. Let $L=lcm(m,n)=\prod\limits_{i=1}^s p_i^{r_i}$. Suppose that $p_i^{r_i}$ does not divide $m$. Is it true that $p_i^{r_i}|n$?

Answer 1

Yes, and you don't need $\gcd(m,n)=1$ for this to be true. You can construct the LCM by taking the highest power of each prime that divides $m$ or $n$ and multiplying them together, so if $p_i^{r_i}$ is the highest power of $p_i$ that divides the LCM, it divides at least one of $m,n$.

Answer 2

Yes. The only way $p_i^{r_i}$ would be a factor of the $lcm$ is if it a factor of $m$ or a factor of $n$. So if it doesn't divide $m$, then it must divide $n$.