1. **State the problem:** We need to find three integer dimensions $x$, $y$, and $z$ of a cuboid such that its surface area is 340 cm². 2. **Formula for surface area of a cuboid:** $$\text{Surface Area} = 2(xy + yz + zx)$$ where $x$, $y$, and $z$ are the lengths of the edges. 3. **Set up the equation:** $$2(xy + yz + zx) = 340$$ Divide both sides by 2: $$\cancel{2}(xy + yz + zx) = \cancel{2}170$$ $$xy + yz + zx = 170$$ 4. **Find integer solutions:** We want positive integers $x$, $y$, $z$ such that $$xy + yz + zx = 170$$ 5. **Trial and error approach:** Start with possible factors of 170 and test combinations. Try $x=5$: $$5y + 5z + yz = 170$$ Rewrite as: $$yz + 5y + 5z = 170$$ Add 25 to both sides to complete the rectangle: $$yz + 5y + 5z + 25 = 170 + 25$$ $$(y + 5)(z + 5) = 195$$ 6. **Factor 195:** $$195 = 1 \times 195, 3 \times 65, 5 \times 39, 13 \times 15$$ 7. **Check each factor pair:** - If $y+5=13$ and $z+5=15$, then $y=8$, $z=10$. 8. **Verify:** $$xy + yz + zx = 5\times8 + 8\times10 + 5\times10 = 40 + 80 + 50 = 170$$ 9. **Answer:** One set of integer dimensions is $\boxed{5, 8, 10}$. These dimensions satisfy the surface area requirement of 340 cm².