Suppose $a_1, a_2, . . . , a_8$ are eight distinct integers from $\{1, 2, . . . , 16, 17\}$. Show that there is an integer $k > 0$ such that there are at least three different (not necessarily disjoint) pairs $(i, j)$ such that $a_i - a_j = k$. Also find a set of seven distinct integers from $\{1, 2, . . . , 16, 17\}$ such that there is no integer $k > 0$ with that property.