Show that: (a) It is possible to divide all positive integers into three groups $A_1$, $A_2$, and $A_3$ such that for every integer $n\ge 15$, it is possible to find two distinct elements whose sum equals $n$ from all of $A_1$, $A_2$, and $A_3$. (b) Dividing all positive integers into four groups, then regardless of the partition, there must exists an integer $n\ge 15$ such that it is impossible to find two distinct numbers whose sum is $n$ in at least one of these four groups.