The product of a set of distinct positive integers is 84. What is the least possible sum of these integers?
To minimize the sum of the integers, I want to minimize the number of factors and make them as close to each other as possible. Since 84 is even, I can start by dividing it by 2 and see what the other factor is. 84 / 2 = 42, so one possible set of factors is {2, 42}. The sum is 44, but I can do better by factoring 42 further. 42 / 2 = 21, so another possible set of factors is {2, 2, 21}. The sum is 25, which is smaller than 44, but I can still do better by factoring 21 further. 21 / 3 = 7, so the best possible set of factors is {2, 2, 3, 7}. The sum is 14, and I cannot do better than this, because any other factorization of 84 would either increase the number of factors or increase the difference between them. Therefore, the least possible sum of the distinct positive integers that multiply to 84 is 14. 