<s>[INST] Least value of $x+y+z$ where $ax=by=cz$ The following question is a generalization of the case $a=3$, $b=4$, $c=5$ from a MindYourDecisions YouTube video (which I am not going to actually link here).

Given positive integers $a$, $b$, and $c$, what is the smallest possible value of $x+y+z$ where $x$, $y$, and $z$ are positive integers with $ax=by=cz$?

If $a$, $b$, and $c$ are pairwise coprime, then the answer is just $bc+ac+ab$, because $lcm(a,b,c)=abc$.
The case $a=3$, $b=4$, $c=7$ was asked 6 years ago in Least Value Of $x+y+z$.
 [/INST] Let $N = ax = by = cz$. Since $N$ is a multiple of $a, b, c$ it must be a multiple of $\mathrm{lcm}(a, b, c)$. Thus, we should take
$$
\begin{align*}
x &= N/a \\
y &= N/b \\
z &= N/c
\end{align*}
$$
The sum will be minimal when $N = \mathrm{lcm}(a, b, c)$.
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