One of the legs of a right-angled triangle is 13, and the other two sides are expressed in integers. find the other two sides.

Suppose the other two sides of a right-angled triangle are x and y.
By the Pythagorean theorem, we can compose the following equation for a given triangle:
x² + 13² = y²,
y² – x² = 13²,
(y-x) * (y + x) = 13².
The number 13 is a prime number, which means that the right side of the equation can have only two variants of meaning:
(y – x) * (y + x) = 13 * 13 or
(y – x) * (y + x) = 169 * 1.
In the first version, we get that
y – x = 13 and y + x = 13, which is possible only if y = 13 and x = 0, but the side of the triangle cannot be 0, so this option is not suitable.
Consider the second option:
y – x = 1,
y + x = 169.
From the first expression we find that y = x + 1, we substitute this value into the second equation:
x + x + 1 = 169,
2 * x = 168,
x = 84.
So y = 84 + 1 = 85.
Answer: 84 and 85.