A right-angled triangular prism has all edge lengths as integers. The prism has faces with areas of 30 and 13. What is the volume of the prism?
Solution. The length of every edge of the cuboid is an integer, which means the sides of the right-angled triangle are Pythagorean triples.

One face of the cuboid has an area of 13. We will show that this face cannot be a right-angled triangle. If it were, by completing the right-angled triangle to a rectangle with sides equal to the two legs, the area of this rectangle would be 26. However, 26 can only be factored into integers as 2 and 13 or 1 and 26, and neither 1 nor 2 is part of a Pythagorean triple.

Therefore, the face with an area of 13 must be one of the rectangular faces forming the sides of the cuboid. $13 = 13 \cdot 1$, and 13 is part of a Pythagorean triple. Thus, 13 is one edge of the base, and 1 is the height of the cuboid. This means that one side of the right-angled triangle is 13.

If one of the rectangular faces has an area of 30, then, since the height of the cuboid is 1, one side of the right-angled triangle is 30. However, 30 and 13 cannot form a Pythagorean triple.

Therefore, the area of the right-angled triangle is 30. Its hypotenuse is 13, because 13 is not a divisor of 30. There is only one Pythagorean triple where the hypotenuse is 13, which is the $5 ; 12 ; 13$ triple, and it satisfies the conditions $\left(5 \cdot \frac{12}{2}=30\right)$.

Although we could have calculated the volume of the cuboid using the height and the area of the right-angled triangle, we will do it now: $1 \cdot 30$ surprisingly equals 30.

Thus, the volume of the cuboid is 30.