Another one.

corrected semi-answer

Alex Y

The sides of the triangles with such legs total to 1528. Answer and rationale deferred...

Answer

Alex Y

There are four such triangles, with sides of 40,42,58; 42,56,70; 42,144,150; and 42,440,442.

If a right tringle has legs of 42 and x, and hypotenuse of y, Pythagoras tells us that 42^2 + x^2 = y^2. A little algebraic manipulation converts this:

42^2 = y^2-x^2

42^2 = (y-x)*(y+x)

Since x ad y are both integers, y-x and y+x are either both odd or both even.

42^2 = 2^2 * 3^2 * 7^2

We are looking for pairs of numbers that multiply to 42^2, and are both odd or both even. The only way that can happen is for them both to be even. The five choices are:

2 * (2*3^2*7^2)

(2*3) * (2*3*7^2)

(2*7) * (2*3^2*7)

(2*3^2) * (2*7^2)

(2*3*7) * (2*3*7)

The last one reduces to a non-triangle, since x = 0

But for the others, you can determine x and y. For instance,

(2*7) * (2*3^2*7)

= 14 * 126

70 is half-way between 14 and 126, and they can be expressed as

(70-56) * (70+56)

so 56 and 70 form the other sides of one of the four triangles.

perfect