1. The problem statement, all variables and given/known data Prove that for every k >= 2 there exists a number with precisely k divisors. I know the solution, but don't fully understand it, here it is; Consider any prime p. Let n = p^(k-1). An integer divides n if and only if it has the form p^i where 0<= i <= (k-1). There are k choices for i, therefore n has exactly k divisors. Could someone fully explain the thought process involved in finding the solution, I understand p^i etc, just don't know where p^(k-1) comes from.