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.