Number theory / Trigonometry  Quant/Math  CAT 2016
Question 4 the day:
June 28, 2002
The question for the day is an interesting number systems / trigonometry problem.
Question:
 'a' and 'b' are the lengths of the base and height of a right angled triangle whose hypotenuse is 'h'. If the values of 'a' and 'b' are positive integers, which of the following cannot be a value of the square of the hypotenuse?

(1) 13
(2) 23
(3) 37
(4) 41
Correct choice (2). Correct Answer  23
Solution:
The value of the square of the hypotenuse = h^{2} = a^{2} + b^{2}
As the problem states that 'a' and 'b' are positive integers, the values of a^{2} and b^{2} will have to be perfect squares. Hence we need to find out that value amongst the four answer choices which cannot be expressed as the sum of two perfect squares.
Choice 1 is 13. 13 = 9 + 4 = 3^{2} + 2^{2}. Therefore, Choice 1 is not the answer as it is a possible value of h^{2}
Choice 2 is 23. 23 cannot be expressed as the sum two numbers, each of which in turn happen to be perfect squares. Therefore, Choice 2 is the answer.
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