Question 31
Let $k \gt 2$ be an integer. We call a positive integer $\ell$ $k$-pable if we can partition the numbers $1, 3, 5, \ldots, 2k-1$ into two sets $A$ and $B$ such that the sum of the elements of $A$ is exactly $\ell$ times as large as the sum of the elements of $B$. Prove that the smallest $k$-pable number is relatively prime to $k$.