My problem seems elementary. However a post in SE 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.

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 $m_1'a_1+m_2'a_2+m_3'a_3=A$.

The motivation for this problem is the following problem I conjecture but can not prove either.

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. $\mathbb Q[x,y,z]_A=\mathrm{span}\big\{x^uy^vz^w:u,v,w\geq0,\;ua+vb+wc=A\big\}$.

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.