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The sum of the squares of two consecutive positive odd numbers is 514. Find the numbers.

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13, 15

11, 13

15, 17

17, 19

answer is B.

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Detailed Solution

Given that the sum of the squares of two consecutive positive odd numbers is 514.
Let's assume that n and

n + 2

are the two consecutive positive odd numbers.
According to the question,

⟹ n^{2} + {(n + 2)}^{2} = 514

⟹ n^{2} + n^{2} + 4 n + 4 = 514

⟹ {2 n}^{2} + 4 n + 4 - 514 = 0

⟹ 2 n^{2} + 4 n - 510 = 0

⟹ n (n + 17) - 15 (n + 17) = 0

⟹ (n - 15) (n + 17) = 0

⟹ n - 15 = 0 or n + 17 = 0

⟹ n = 15 or n = - 15

(not possible)

⟹ n = 15

∴ n + 2 = 15 + 2 = 17

Thus, the integers are 15 and 17.
Hence, option (2) is correct.

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