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Question

The sum of all positive integers n for which 13+23....+(2n)312+22+...n2 is also an integer is.


A

8

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B

9

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C

-2

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D

-4

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Solution

The correct option is A

8


13+23....+(2n)312+22+...n2=(2n(2n+1)22)26n(n+1)(2n+1)

=6n(2n+1)n+1

=12n2+6nn+1=12(n2+1)+6(n+1)+6n+1

=12n6+6n+1

If the given terms are an integers, then 6n+1 must be an integer

n=1,2,5

Sum=8


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