Problem of the Month
Problem 1: October 2023

Given a positive integer $n$, the digit sum of $n$ is the sum of the base $10$ digits of $n$. We will denote the digit sum of $n$ by $D(n)$. For example, $D(1409)=1+4+0+9=14$.

Suppose that $m$ is a positive integer. We will call a list of consecutive positive integers $$a,a+1,a+2,\dots ,a+k$$ an $m$-list if none of $D(a)$, $D(a+1)$, $d(a+2)$, and so on up to $D(a+k)$ is a multiple of $m$. For example, the list $997,998,999,1000,1001,1002$ is a $4$-list because the digit sums of the integers in the list are $25$, $26$, $27$, $1$, $2$, and $3$, respectively, none of which is a multiple of $4$.

This problem explores the maximum length of an $m$-list for a few values of $m$.

1. Show that the maximum length of a $2$-list is $2$. To do this, you must show that there is a $2$-list of length $2$ and you must also show that no list of three or more consecutive positive integers can be a $2$-list.

2. Show that the maximum length of a $7$-list is $12$.

3. Determine the maximum length of a $9$-list.

4. Determine the maximum length of an $11$-list.