Question 115
Determine the smallest positive integer $q$ with the following property:
for every integer $m$ with $1 \leqslant m \leqslant 1006$, there exists an integer $n$ such that
$$
\frac{m}{1007} q \lt n \lt \frac{m+1}{1008} q .
$$
for every integer $m$ with $1 \leqslant m \leqslant 1006$, there exists an integer $n$ such that
$$
\frac{m}{1007} q \lt n \lt \frac{m+1}{1008} q .
$$