RU-ALLRUSSIAN 1988 Problem 23

The sequence of integers $\mathrm{a}_{\mathrm{n}}$ is given by $\mathrm{a}_{0} = 0$ , $\mathrm{a}_{\mathrm{n}} = \mathrm{p}(\mathrm{a}_{\mathrm{n} - 1})$ , where $\mathrm{p}(\mathrm{x})$ is a polynomial whose coefficients are all positive integers. Show that for any two positive integers $\mathrm{m}$ , $\mathrm{k}$ with greatest common divisor d, the greatest common divisor of $\mathrm{a}_{\mathrm{m}}$ and $\mathrm{a}_{\mathrm{k}}$ is $\mathrm{a}_{\mathrm{d}}$ .