22. The base of a right prism is a right triangle. All edges of this prism have natural lengths, and the areas of some two of its faces are 13 and 30. Find the sides of the base of this prism.
- Answer: 5, 12, 13. We will prove that at least one of the two specified faces is a base. Suppose this is not the case. If the height is 1, then the sides of the triangle at the base are 13 and 30, which cannot be the sides of a right triangle with natural sides. If the height of the prism is greater than one, then the areas of the lateral faces are divisible by this height and cannot be coprime. Therefore, either 13 or 30 is the area of the base. If the area of the base is 13, then the product of the legs is 26, which means it is either 1 and 26, or 2 and 13, in both cases the hypotenuse does not turn out to be a natural number. Therefore, the area of the base is 30, and the area of one of the lateral faces is 13. This can only be if the height is 1, and the side of the base is 13 (a side of a right triangle with natural sides cannot be 1). The product of the legs is 60, so there is no side equal to 13 among them, which means 13 is the hypotenuse. Then it is easy to verify that the legs are 12 and 5.

(KöMaL journal)