Putnam 2010 A1

Given a positive integer $n$, what is the largest $k$ such that the numbers $1,2,\dots,n$ can be put into $k$ boxes so that the sum of the numbers in each box is the same? [When $n=8$, the example $\{1,2,3,6\}, \{4,8\}, \{5,7\}$ shows that the largest $k$ is at least 3.]