KVPY SX SB Exam 2015 Question Paper

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Question : 3 of 120

Marks: +1, -0

The largest perfect square that divides

2014^3- 2013^3 + 2012^3- 2011^3

+ .... + 2^3- 1^3

is -

1 $^{2}$
2 $^{2}$
1007 $^{2}$
2014 $^{2}$

Solution:

2{\{(2014)^3 + (2012) ^2 +…+ 2^3 }

– {(2014)^3 + (2013)^3 +…+ 1^3 \}

= 2 × 8 {\{(1007)^2 + (1006)^2 +…+ 1^3 \}

- {\{(2014)^3 + (2013)^2 +…+ 1^3 \}}

= 2 × 8 × ({(1007)(1008)}/2)^2

-({(2014)(2015)}/2)^2

= 2 × 8 ×{(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

(1007)^2

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