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The largest perfect square that divides ...

The largest perfect square that divides `2014^(3)-2013^(3)+2012^(3)-2011^(3)+ . . . +2^(3)-1^(3)` is

A

`1^(2)`

B

`2^(2)`

C

`1007^(2)`

D

`2014^(2)`

Text Solution

Verified by Experts

The correct Answer is:
C
`2{(2014)^(3)+(2012)^(2)_ . . . +}-{(2014)^(3)+(2013)^(3)+ . . .1^(3)}`
`=2xx8{(1007)^(2)+(1006)^(2)+ . . . +1^(3)}-{(2014)^(3)+(2013)^(3)+ . . .+1^(3)}`
`=2xx8xx(((1007)(1008))/(2))^(2)-(((2014)(2015))/(2))^(2)`
`=2xx8xx((1007)^(2)(1008)^(2))/(4)-((2014)^(2)(2015)^(2))/(4)`
`=(1007)^(2)(2016)^(2)-(1007)^(2)(2015)^(2)`
`(1007)^(2){2016-2015}{2016+2015}`
`(1007)^(2)(4031)`
=divisible by `(2007)^(2)`
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