The sum of all positive integers n for which `(1^3+2^3+....(2n)^3)/(1^2+2^2+....+n^2)` is also an integer is

Options

A. 8
B. 9
C. 15
D. Infinite

Answer

A

Explanation

`((2n(2n+1))/2)^(2)/((n(n+1)2n+1)/6)=(6n^(2)(2n+1)^(2))/(n(n+1)(2n+1))`
`=(6n(2n+1))/((n+1))=(6n(n+1+n))/(n+1)`
`6n+(6n^(2))/(n+1)=6n+6(n-1)+6/(n+1)="Integer"`
The value of 'n' which are satisfying are n=1,2,5 only for being integer
`:. " sum "=1+2+5=8`