I recently came across the question:
A nine-digit number $x_1$$x_2\cdots x_9$ is formed using each of the digits $1,2,3,...,9$ exactly once. Reading left to right for each $x_1\cdots x_i$, i divides that number exactly. Find the number.
Stated in another way:
$$x_1x_2 \equiv 0 \mod 2$$
$$x_1x_2x_3 \equiv 0 \mod 3$$
$$\cdots$$
$$x_1x_2\cdots x_9 \equiv 0 \mod 9$$
I solved this question in a very similar way to solution presented here even getting this intermediate form
$$x_1 4 x_3 2 5 8 x_7 6 x_9$$ $$y_1 8 y_3 6 5 4 y_7 2 y_9$$
and then I tried out the possibilities until I landed on the solution.
My solution and many of the ones I have seen feel very ad-hoc with lots of trial and error. I was wondering if there were more direct and/or elegant solutions (especially using modular arithmetic).