parmenides51 wrote: Prove that if $p$ is an odd prime, then $p^2(p^2 -1999)$ is divisible by $6$ but not by $12$. We want to prove that: $p^2(p^2-7)$ is divisible by 6 but not by 12.(Since $p^2(p^2 -1999)=p^2(p^2-7)-1992p^2$ and $1992$ is divisible by 12) First, we will prove that $p^2(p^2-7)$ is divisible by 6, or $p^4-p^2$ is divisible by 6. Since $p^4-p^2=p(p-1)p(p+1), 3|(p-1)p(p+1), 2|p(p+1), (3,2)=1$, we get the desired result. Now, $p^2(p^2-7)$ is not divisible by 12. Since $p$ is an odd prime, $p^2\equiv1\mod4$ so $p^2(p^2-7)\equiv1(1-7)=2\mod4$ So $p^2(p^2-7)$ is not divisible by 4, which leads to the desired result.