My problem seems elementary. However a post in [SE](https://math.stackexchange.com/questions/4304278/least-common-multiple-of-three-integers) has not got an answer.

Let $a\_1,a\_2,a\_3\geq1$ be integers and let $A=\mathrm{lcm}(a\_1,a\_2,a\_3)$ be their least common multiple. I want to show the following.

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> If $m\_1,m\_2,m\_3\geq0$ are natural numbers satisfying $m\_1a\_1+m\_2a\_2+m\_3a\_3=2A$, then there exist integers $m\_1',m\_2',m\_3'$ satifying $0\leq m\_i'\leq m\_i$ for each $i$ and $m\_1'a\_1+m\_2'a\_2+m\_3'a\_3=A$.
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The motivation for this problem is the following problem I conjecture but can not prove either.

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> Let $\mathbb Q[x,y,z]$ be the polynomial ring of three variables. Let $a,b,c\geq1$ be the degrees of $x,y,z$ respectively. Let $A=\mathrm{lcm}(a,b,c)$. Then $\mathbb Q[x,y,z]\_{(A)}:=\bigoplus\_{d\geq0}\mathbb Q[x,y,z]\_{Ad}$ is generated by $\mathbb Q[x,y,z]\_A$. The subscript in $\mathbb Q[x,y,z]\_A$ denotes the degree $A$ subspace, i.e. $\mathbb Q[x,y,z]\_A=\mathrm{span}\big\{x^uy^vz^w:u,v,w\geq0,\;ua+vb+wc=A\big\}$.
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The first statement will prove the second. Consider $x^uy^vz^w\in\mathbb Q[x,y,z]\_{Ad}$ with $ua+vb+wc=Ad$.

* It suffices to show that there exist $\begin{cases}0\leq u'\leq u\\0\leq v'\leq v\\0\leq w'\leq w\end{cases}$ and $u'a+v'b+w'c=A$. This is because we then have $x^uy^vz^w=\big(x^{u'}y^{v'}z^{w'}\big)\big(x^{u-u'}y^{v-v'}z^{w-w'}\big)\in\mathbb Q[x,y,z]\_A\cdot\mathbb Q[x,y,z]\_{A(d-1)}$, and we reduce $d$ to $d-1$.
* Also if $ua\geq A$ (or $vb\geq A$ or $wc\geq A$), we just take $\begin{cases}u'=A/a\\v'=0\\w'=0\end{cases}$ as a solution.
* So the difficult situation is when $\begin{cases}0\leq u<A/a\\0\leq v<A/b\\0\leq w<A/c\end{cases}$. In this case $dA=ua+vb+wc<3A$, so $d<3$. This reduces to the $d=2$ case, which is the first statement.

Thanks for any comments.