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only one pair of positive integers

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only one pair of positive integers

by sanju09 » Mon Mar 07, 2011 12:20 am
If p < q, then 3^2 + 4^2 + 5^2 + 12^2 = p^2 + q^2 is satisfied by only one pair of positive integers (p, q). What is the value of p + q?
(A) 24
(B) 20
(C) 18
(D) 15
(E) 14


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by Anurag@Gurome » Mon Mar 07, 2011 1:03 am
sanju09 wrote:If p < q, then 3^2 + 4^2 + 5^2 + 12^2 = p^2 + q^2 is satisfied by only one pair of positive integers (p, q). What is the value of p + q?
(A) 24
(B) 20
(C) 18
(D) 15
(E) 14
Easiest way to solve this problem is to exploit the knowledge that (3, 4, 5) and (5, 12, 13) are Pythagorean triplets.

Hence, 3² + 4² = 5² and 5² + 12² = 13²

As there only one such pair, p² + q² = 5² + 13²
Therefore, (p + q) = (5 + 13) = 18

The correct answer is C.
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