Can anyone help me with this maths problem?

For a, b, and k ALL to be positive integers, the smallest possible value of k = 10.

When k = 10, the smallest possible values of a and b are

a = 69 and b = 20.

This will result in a right triangle with sides of 585 and 928, with a hypotenuse of 1097.

How do you do it?

Using the Pythagorean Theorem:

(5a + 12b)² + (12a + 5b)² = (13a + kb)²

This can be reduced to:

240a + 169b = k(26a + kb)

When k is a positive integer less than 10, it is impossible that both a and b are positive. (Try it.)

When k = 10, we are left with:

240a + 169b = 260a + 100b

This can be simplified to:

20a = 69b

You can see that is it now possible that both a and b can be positive.

The lowest number where 69b can be a multiple of 20...when b is a positive integer...is 1380. When 69b = 1380, b = 20.

And...when b = 20, a = 69.

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