I OM - B - Task 20

The sides of a right-angled triangle are expressed by natural numbers. One of the legs is expressed by the number 10. Calculate the remaining sides of this triangle.
Natural numbers $ x $ and $ y $ representing the length of the hypotenuse and the length of the other leg of the considered triangle satisfy, according to the Pythagorean theorem, the equation

Since $ 10^2 $ is an even number, it follows from equation (1) that the numbers $ x^2 $ and $ y^2 $ are either both even or both odd. Therefore, the numbers $ x $ and $ y $ are either both even or both odd.
From equation (1) it follows

Since the numbers $ x $ and $ y $ are of the same parity, the numbers $ (x + y) $ and $ (x-y) $ are both even; moreover, $ x + y > x-y $. The number $ 10^2 = 100 $ can be factored into the product of two unequal even factors in only one way: $ 100 = 50 \cdot 2 $. Therefore, from equation (2) it follows that

Thus $ x = 26 $, $ y = 24 $.