# math

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Let 5a + 12b and 12a + 5b be the side lengths of a right-angled triangle and 13a + kb be the hypotenuse, where a, b and k are positive integers. Find the smallest possible value of k and the smallest values of a and b for that k

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Using Pythagoras, we get
(5a+12b)2+(12a+5b)2 = (13a+kb)2

Solving for k:
k=(sqrt(169*b^2+240*a*b+169*a^2)-13*a)/b
or
k=-(sqrt(169*b^2+240*a*b+169*a^2)+13*a)/b
The second solution is negative and does not fit the requirements that k is a positive integer.

We are required to find pairs of positive integers a and b such that k is also a positive integer.
I have not been able to find an analytical solution of the integer problem, although the minimum value of k appears to be ten, with the minimum values of a and b being 69 and 20.
Thus the triplet (a,b,k) of (69,20,10) is the minimal solution.
The solution for k=11 is (24,23,11), and for k=12, (25,72,12). Other solutions exist for the same k where a and b are multiples of the basic case.

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maths... keewwllllll