I cannot tackle the problem.
Let $R$ be a commutative ring with unity. Let $a$, $b$, $c\in R$ be such that there exists $x$, $y$, $z\in R$ with $$xa+yb+zc=1.$$ Show that there exist $\alpha$, $\beta$, $\gamma\in R$ such that $$\alpha a^{15} +\beta b^{16}+\gamma c^{17}=1.$$
Can anyone give a hint how to start? Actually I was thinking about binomial expansion. Will it be helpful or any other way would be better?