34. Find the smallest positive integer $n$ such that $n(n+1)(n+2)$ is divisible by 247 .
34. Ans: 37 .
Since $247=13 \cdot 19$, one of $n, n+1, n+2$ is divisible by 13 , call it $a$ and one by 19 call it $b$. Clearly $|b-a| \leq 2$.
Let $b=19 c$. When $c=1$, since $|b-a| \leq 2, a$ is among $17,18,19,20,21$. But none is divisible by 13 , hence we try $c=2$. Now $b=38$ and $a$ is among $36,37,38,39,40$, hence $a=39$. Thus the least $n=37$.