1. **Problem statement:** We have a cuboid with three visible faces having areas 12, 15, and 20 square meters. The edges of the cuboid are integers. We need to find the volume of the cuboid. 2. **Known information:** Let the edges of the cuboid be $x$, $y$, and $z$ (all integers). 3. **Face areas:** The areas correspond to the products of pairs of edges: - $xy = 12$ - $yz = 15$ - $xz = 20$ 4. **Goal:** Find the volume $V = xyz$. 5. **Method:** Multiply all three equations: $$ (xy)(yz)(xz) = 12 \times 15 \times 20 $$ $$ (xyz)^2 = 3600 $$ 6. **Solve for $xyz$:** $$ xyz = \sqrt{3600} = 60 $$ 7. **Check integer edges:** Since $xyz=60$ and the face areas are products of integer edges, the edges $x$, $y$, and $z$ must be integers satisfying the given face areas. 8. **Find edges:** From $xy=12$, possible integer pairs are $(1,12), (2,6), (3,4)$. From $yz=15$, possible pairs are $(1,15), (3,5), (5,3), (15,1)$. From $xz=20$, possible pairs are $(1,20), (2,10), (4,5), (5,4), (10,2), (20,1)$. 9. **Find common edges:** Try $x=3$, $y=4$, then $xy=12$ correct. Check $yz=15$ with $y=4$: $4z=15$ no integer $z$. Try $x=4$, $y=3$, $xy=12$ correct. Check $yz=15$ with $y=3$: $3z=15$ so $z=5$ integer. Check $xz=20$ with $x=4$, $z=5$: $4 \times 5=20$ correct. 10. **Edges are $x=4$, $y=3$, $z=5$ and volume is:** $$ V = 4 \times 3 \times 5 = 60 $$ **Final answer:** The volume of the cuboid is **60** cubic meters.