7. Let's call a natural number useful if it does not contain any zeros or identical digits in its decimal representation, and the product of all its digits is divisible by their sum. Find the two largest consecutive (i.e., differing by 1) useful numbers.
Solution. The numbers 9875213 and 9875214 are useful. We will prove that there do not exist larger consecutive useful numbers. The sums of the digits of consecutive numbers are also consecutive (otherwise, we would have a transition across a place value, meaning a 0 at the end). However, the maximum possible sums of the digits are 35 and 36 (among any two larger sums, at least one has a prime divisor greater than 9, or exceeds $1+\ldots+9=45$). Consecutive useful numbers cannot have more than seven digits, otherwise, the sums of their digits would be at least $1+\ldots+8=36$ for each. Consecutive useful numbers of the form 9876 * $* *$ do not exist, as the sums of their digits are at least $9+8+7+6+1+2+3=36$. Consecutive useful numbers of the form $9875 * * *$ can only differ from the found ones by the permutation of the last digits (otherwise, the sums of their digits would be greater than 36), so the found numbers are indeed the largest. Criteria. 5 points are given for the estimation, and 2 points for the example.