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Quadratic Equations

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Quadratic Equations

by harsh.champ » Tue Feb 09, 2010 5:10 am
It is known that the largest integer n such that each prime factor of n (n + 1) is = 11 is n = 9800. What is the largest integer y having each prime factor = 11 and satisfying the equation x^2 - 2y^2 = 1 for some integer x?

13860
13001
12761
11231
None of these

The OA is A.
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by shashank.ism » Tue Feb 09, 2010 5:17 am
harsh.champ wrote:It is known that the largest integer n such that each prime factor of n (n + 1) is = 11 is n = 9800. What is the largest integer y having each prime factor = 11 and satisfying the equation x^2 - 2y^2 = 1 for some integer x?

13860
13001
12761
11231
None of these

The OA is A.
The equation x2 - 2y2 = 1 implies that x is odd. Write x = 2n + 1. Then
2y2 = (2n + 1)2 -1 = 4n2 + 4n = 4n(n + 1):
Since y has all of its prime factors = 11, so does n (n+1).
--> y2 = 2n (n + 1) = 2 x 9800 x 9801 = 24 x 34 x 52 x 72 x 112:

Hence, y = 22x32x5x7x11 = 13860. With y = 13860 and x = 2n+1 = 2x9800+1 = 19601,
it is easy to see that x2 - 2y2 = 1 is in fact satisfied. Therefore the answer is 13860.
Ans A
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