1. **Problem statement:** The product of two whole numbers is 72. We need to find the largest possible sum $A$ and the smallest possible sum $B$ of these two numbers, then calculate $A - B$. 2. **Understanding the problem:** We want two whole numbers $x$ and $y$ such that: $$xy = 72$$ and we want to find: $$A = \max(x + y)$$ $$B = \min(x + y)$$ 3. **Step 1: Find all pairs of whole numbers whose product is 72.** Since the set of whole numbers includes 0, 1, 2, 3, ..., we consider pairs $(x,y)$ with $x,y \geq 0$ and $xy=72$. 4. **Step 2: List factor pairs of 72:** $$ (1,72), (2,36), (3,24), (4,18), (6,12), (8,9) $$ Note: 0 cannot be a factor because $0 \times y = 0 \neq 72$. 5. **Step 3: Calculate sums for each pair:** $$1 + 72 = 73$$ $$2 + 36 = 38$$ $$3 + 24 = 27$$ $$4 + 18 = 22$$ $$6 + 12 = 18$$ $$8 + 9 = 17$$ 6. **Step 4: Identify largest and smallest sums:** $$A = 73$$ (largest sum) $$B = 17$$ (smallest sum) 7. **Step 5: Calculate $A - B$:** $$A - B = 73 - 17 = 56$$ **Final answer:** $$\boxed{56}$$