Let $a_1,a_2,a_3\in\mathbb Z_{\geq1}$ be three integers and let $A$ be their least common multiple. I want to prove the following.
If $m_1,m_2,m_3\in\mathbb Z_{\geq0}$ such that $m_1a_1+m_2a_2+m_3a_3=2A$, then there exists $m_1',m_2',m_3'\in\mathbb Z_{\geq0}$ such that each $m_i'\leq m_i$ and $m_1'a_1+m_2'a_2+m_3'a_3=A$.
A weaker but seemingly more meaningful result is the following, which I can not prove either.
If $\mathbb Q[x,y,z]$ is a polynomial ring, and $x,y,z$ have their degrees $a,b,c\geq1$ respectively, then $\mathbb Q[x,y,z]_A$ generate $\mathbb Q[x,y,z]_{(A)}$ for $A=\mathrm{lcm}(a,b,c)$, where $\mathbb Q[x,y,z]_A$ is spanned by $x^uy^vz^w$ for all $u,v,w\in\mathbb Z_{\geq0}$ such that $ua+vb+wc=A$, and $\displaystyle \mathbb Q[x,y,z]_{(A)}=\bigoplus_{d\geq 0}\mathbb Q[x,y,z]_{Ad}$.
Otherwise, counter-examples are welcome.