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InterviewSolution
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The largest perfect square that divides `2014^(3)-2013^(3)+2012^(3)-2011^(3)+ . . . +2^(3)-1^(3)` isA. `1^(2)`B. `2^(2)`C. `1007^(2)`D. `2014^(2)` |
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Answer» Correct Answer - 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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