In decimal representation, we call an integer k-carboxylic if and only if it can be represented as a sum of k distinct integers, all of them greater than 9, whose digits are the same. For instance, 2008 is 5-carboxylic because $2008 = 1111 + 666 + 99 + 88 + 44$. Find, with an example, the smallest integer k such that 8002 is k-carboxylic.
I started by writing $8002 = 1111a + 111b + 11c$. To minimize k, we want to maximize a, then b and c. But it seems there are still a lot of combinations. Help?