Let $R$ be a commutative ring with identity. Let $a,b,c\in R$ such that $\exists $ $x,y,z\in R$ that satisfy $ax+by+cz=1$.

Prove that there exist $\alpha,\beta,\gamma\in R$ such that 

\begin{equation\*}
\alpha a^{15}+\beta b^{16}+\gamma c^{17}=1.
\end{equation\*}

I have not made any progress in this problem as I don't know how to start it. Please help.