## Task 2 - 180932

In the task of the 2nd level, it was to be shown that among four consecutive six-digit numbers, it is not necessarily the case that one must have a digit sum divisible by 4.

Determine the largest natural number $n$ for which the following statement is true:

"There are $n$ consecutive natural numbers, among which there is none whose digit sum is divisible by 4."
The sought number $n$ is 6. To show this, we first demonstrate that there are 6 consecutive natural numbers whose digit sums are all not divisible by 4, and then that for any 7 consecutive natural numbers, at least one of them has a digit sum that is divisible by 4.

We consider the natural numbers 4997, 4998, 4999, 5000, 5001, and 5002. These 6 consecutive natural numbers have the digit sums 29, 30, 31, 5, 6, and 7, none of which are divisible by 4.

Seven consecutive natural numbers can span at most two "decades," or intervals of the form $[k \cdot 10+0 ; k \cdot 10+9]$ with non-negative integer $k$. Then, in at least one such interval, there must be 4 consecutive numbers whose digit sums are 4 consecutive natural numbers, one of which is divisible by 4.