HMMT February

The second condition implies that 16 divides $a(2 a-1)\left(2 a^{2}-a-1\right)$, which shows that $a \equiv 0$ or 1 modulo 16. The case $a=1$ would contradict the triviality-avoiding condition $N>1$. $a$ cannot be 16, because 7 does not divide $a(2 a-1)\left(2 a^{2}-a-1\right)$. a cannot be 17, because 9 does not divide $a(2 a-1)\left(2 a^{2}-a-1\right)$. It can be directly verified that $a=32$ is the smallest positive integer for which $1+2+\cdots+(N-1)=2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13 \cdot 31$ which is divisible by $1,2, \ldots, 10$. For this $a$, we compute $N=32(2 \cdot 32-1)=2016$.