Digital sum device!

We'll start with 10.

If a number is divisible by 10, it is divisible by 2 and 5 also.

If a number is divisible by 9, it is divisible by 3 too.

If a number is divisible by 8, it is divisible by 4 too.

And since the number is now divisible by 2 and 3, it's divisible by 6 as well.

So what we're looking for is the LCM (least common multiple) of 7, 9, 8, 10 = 7×9×8×5 = 2520.

Every given between 1 and 10 has prime factorization of the form $2^{\alpha_2}3^{\alpha_3}5^{\alpha_5}7^{\alpha_7}$ where. Since $n$ is the least common multiple of all these integers, $n=2^{\max(\alpha_2)}3^{\max(\alpha_3)}5^{\max(\alpha_5)}7^{\max(\alpha_7)}=2^33^25^17^1=2520$ and $2+5+2+0=\boxed{9}$ .