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Question 1093005: A fraction of the form 1/n (n is an element of a real number, where n is greater than 1), is called a unit fraction. Find the largest two perfect square unit fractions whose sum is also a perfect square rational number.
A) 1/49 and 1/576 B) 1/64 and 1/225 C) 1/9 and 1/16 D)1/25 and 1/144 E) 1/400 and 1/441
Answer by greenestamps(13200)   (Show Source):
You can put this solution on YOUR website!
The sum of ANY two perfect square unit fractions is a rational number:
The denominator of that combined fraction is a perfect square, so the whole fraction is a perfect square rational number if the numerator is a perfect square. So the sum of two perfect square unit fractions is a perfect square rational number if a^2+b^2 is a perfect square -- i.e., if
You should recognize that as the Pythagorean Theorem. So we are looking for integers a, b, and c that can be the sides of a right triangle.
In the problem, we are looking for the LARGEST perfect square unit fractions for which the sum of the two fractions is again a perfect square fraction. If we want the largest fractions, then we want the fractions with the SMALLEST denominators.
Since (3,4,5) is the smallest set of three integers for which a^2+b^2 = c^2, we want the denominators of our two perfect square unit fractions to be 3^2 and 4^2.
So the answer to the problem is that the two largest perfect square unit fractions for which the sum of them is a perfect square rational number are 1/9 and 1/16 -- answer C.
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