Question

A right triangle with integer leg lengths is called cool if the number of square units in its area is equal to three times the number of units in the sum of the lengths of its legs. What is the sum of all the different possible areas of cool right triangles?

Answer 1

Let the integer legs be a and b. The area condition gives

(1/2)ab = 3(a + b) => ab = 6(a + b) => ab - 6a - 6b = 0.

Add 36 to both sides and factor:

(a - 6)(b - 6) = 36.

So a - 6 and b - 6 are positive divisors of 36. Up to order the divisor pairs give (a,b) = (7,42), (8,24), (9,18), (10,15), (12,12). Their areas (1/2)ab are 147, 96, 81, 75, 72. The sum of these distinct areas is

147 + 96 + 81 + 75 + 72 = 471.