Counting Up = Counting Down

At time $t$ , Hansel will say $1 + n t$ will Gretel will say $1000 - m t$ .
Thus, we're interested in the values of $t$ such that

$1 + n t = 1000 - mt \Rightarrow (n+m) t = 999$

So, $t$ must be a positive divisor of $999 = 3 ^3 \times 37$ . There are $(3+1)(1+1) = 8$ of them, namely $1, 3, 9, 27, 37, 111, 333, 999$ .

However, we also require that $n$ and $m$ are both positive integers, which means that $n + m \geq 2$ . Hence, $t \leq \frac{999}{2}$ , which means that we cannot include $t = 999$ as an answer.

For each of remaining 7 values, we have $n = 1, m = \frac{999}{t} - 1$ as a solution set. Thus, there are 7 possible values for $t$ .