# 2007 iTest Problems/Problem 30

## Problem

While working with some data for the Iowa City Hospital, James got up to get a drink of water. When he returned, his computer displayed the “blue screen of death” (it had crashed). While rebooting his computer, James remembered that he was nearly done with his calculations since the last time he saved his data. He also kicked himself for not saving before he got up from his desk. He had computed three positive integers $a, b$, and $c$, and recalled that their product is $24$, but he didn’t remember the values of the three integers themselves. What he really needed was their sum. He knows that the sum is an even two-digit integer less than $25$ with fewer than $6$ divisors. Help James by computing $a+b+c$.

## Solution

Since the prime factorization of 24 is $2^3 \cdot 3$, there are only a few options that need to be tested. WLOG, let $a \le b \le c$, so the possible options are $(1,1,24)$, $(1,2,12)$, $(1,3,8)$, $(1,4,6)$, $(2,2,6)$, and $(2,3,4)$.

Two of the conditions are that the sum is a two digit even number and the sum is less than $25$. The only options left are $(1,3,8)$ with sum as $12$ and $(2,2,6)$ with sum as $10$.

The last condition is that the number has less than $6$ divisors. Since $12$ has $6$ divisors and $10$ has $4$ divisors, James’s desired sum is $\boxed{10}$.