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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k=72/13
To find the smallest possible value of k and the corresponding values of a and b, we can use the Pythagorean theorem.
The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
Given that 5a + 12b and 12a + 5b are the side lengths of the triangle and 13a + kb is the hypotenuse, we can write the equation:
(5a + 12b)^2 + (12a + 5b)^2 = (13a + kb)^2
Expanding and simplifying this equation, we get:
25a^2 + 144ab + 144b^2 + 144a^2 + 25b^2 + 24ab = 169a^2 + 26akb + k^2b^2
Combining like terms, we can rewrite this equation as:
193a^2 + 168ab + 119b^2 = 26akb + k^2b^2
Since we are looking for the smallest possible value of k, we can set a and b to be equal to 1 and substitute them into the equation:
193(1)^2 + 168(1)(1) + 119(1)^2 = 26k(1) + k^2(1)^2
Simplifying this further:
193 + 168 + 119 = 26k + k^2
480 = 26k + k^2
Rearranging the equation:
k^2 + 26k - 480 = 0
Factoring the equation, we find:
(k + 40)(k - 12) = 0
This gives us two possible values for k: -40 and 12. Since k must be a positive integer, the smallest possible value of k is 12.
To find the corresponding values of a and b for k = 12, we can substitute k = 12 back into the original equations:
5a + 12b = 13a + 12b
12a + 5b = 12a + 12b
Simplifying these equations, we get:
5a = 13a
5b = 12b
From the first equation, we find that a = 0. Similarly, from the second equation, we find that b = 0.
However, since the problem states that a and b must be positive integers, the smallest possible values of a and b are 1.
Therefore, the smallest possible value of k is 12, and the corresponding values of a and b are 1.