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 \(m\).

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.

Show that the maximum length of a \(7\)-list is \(12\).

Determine the maximum length of a \(9\)-list.

Determine the maximum length of an \(11\)-list.