Shared Wall Lengths Cfd4Bf

1. **State the problem:** We have two rectangular closets sharing one wall. The linen closet has an area of $12$ ft$^2$ and the clothes closet has an area of $18$ ft$^2$. We need to find all possible whole-number lengths for the shared wall. 2. **Understand the problem:** If the closets share a wall, that wall length must be a side length of both rectangles. Let the shared wall length be $x$ (a whole number). Then the other side lengths are $\frac{12}{x}$ for the linen closet and $\frac{18}{x}$ for the clothes closet. 3. **Conditions:** Since the closets are rectangular, both $\frac{12}{x}$ and $\frac{18}{x}$ must be whole numbers (integers) because side lengths are positive whole numbers. 4. **Find divisors:** The shared wall length $x$ must be a common divisor of both $12$ and $18$. 5. **Find divisors of 12:** $1, 2, 3, 4, 6, 12$ 6. **Find divisors of 18:** $1, 2, 3, 6, 9, 18$ 7. **Find common divisors:** $1, 2, 3, 6$ 8. **Conclusion:** The possible whole-number lengths for the shared wall are $\boxed{1, 2, 3, \text{ and } 6}$ feet.