EE-MO 1999

Prove that if $p$ is an odd prime, then $p^2(p^2 -1999)$ is divisible by $6$ but not by $12$ .

It is known that the quadratic equations $x^2 + 6x + 4a = 0$ and $x^2 + 2bx - 12 = 0$ have a common solution. Prove that then there is a common solution to the quadratic equations $x^2 + 9x + 9a = 0$ and $x^2 + 3bx - 27 = 0$ .