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.
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?