if m,n are positive integers such that; $3m+n=3 lcm(m,n)+gcd(m,n)$

Question

if m,n are positive integers such that; $3m+n=3 lcm(m,n)+gcd(m,n)$ prove that $n|m$

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if $m=nk$ the result follows directly as $\gcd(m,n)=n$ and $lcm(m,n)=m$.

let $\gcd(m,n)=d$ then $$3m+n=\frac{mn}{d}+d$$

But i dont see any thing helpful from here.

Also i think the following inequalities may be helpful :

$\gcd(m,n)\le \min(m,n)$ and

$lcm(m,n)|\le mn$

Answer 1

let $d=\gcd(m,n)$. Let $m=dk,n=dl$, then dividing both sides of the equation $3m+n=3\cdot\mathrm{lcm}(m,n)+\gcd(m,n)$ by $d$ we get $$3k+l=3kl+1\iff (3k-1)(l-1)=0$$ which is true if either $k=\frac13$ or $l=1$. Now $k$ is an integer and cannoot be $\frac13$. So we must have $l=1$, or that $n=d$.