1. **Problem Statement:** We want to find how many possible values there are for the sum $a + b + c$ where $a$, $b$, and $c$ are positive integers such that their product $abc = 72$. 2. **Understanding the problem:** Since $a$, $b$, and $c$ are positive integers and $abc=72$, we need to find all triples $(a,b,c)$ with positive integers that multiply to 72. 3. **Prime factorization:** First, factorize 72 into primes: $$72 = 2^3 \times 3^2$$ 4. **Finding triples:** Each of $a$, $b$, and $c$ can be expressed as products of powers of 2 and 3 such that the exponents add up to 3 for 2's and 2 for 3's across $a$, $b$, and $c$. 5. **Enumerate all triples:** We find all positive integer triples $(a,b,c)$ with $abc=72$ by distributing the prime factors among $a$, $b$, and $c$. 6. **Calculate sums:** For each triple, calculate $a+b+c$. 7. **Count distinct sums:** Count how many distinct sums appear. **Step-by-step approach:** - List all factor triples of 72. - For each triple, compute the sum. - Collect all sums and count unique values. **Factor triples of 72 (positive integers):** - (1,1,72) sum = 74 - (1,2,36) sum = 39 - (1,3,24) sum = 28 - (1,4,18) sum = 23 - (1,6,12) sum = 19 - (1,8,9) sum = 18 - (2,2,18) sum = 22 - (2,3,12) sum = 17 - (2,4,9) sum = 15 - (2,6,6) sum = 14 - (3,3,8) sum = 14 - (3,4,6) sum = 13 **Unique sums:** 74, 39, 28, 23, 19, 18, 22, 17, 15, 14, 13 Count of unique sums = 11 **Final answer:** There are **11** possible values for the sum $a+b+c$ when $a,b,c$ are positive integers with $abc=72$.