ILL 1988 Problem 10

Let $a$ be the greatest positive root of the equation $x^3 - 3 \cdot x^2 + 1 = 0.$ Show that $\left[a^{1788} \right]$ and $\left[a^{1988} \right]$ are both divisible by 17. Here $[x]$ denotes the integer part of $x.$