Q. 254.4( 16 Votes )

Prove that n2 – n is divisible by 2 for every positive integer n.

Answer :

Case I: Let n be an even positive integer.


Then, n = 2q


we have


n2 – n = (2q)2 – 2q


= 4q2 – 2q


= 2q (2q – 1 )


n2 – n = 2r, where r = q (2q – 1)


n2 – n is divisible by 2 .


Case: II


Let n be an odd positive integer.


Then, n = 2q + 1


n2 –n


= (2q + 1)2 – (2q + 1)


Taking (2q+1) common from both the terms


= (2q +1) (2q+1 –1)


= 2q (2q + 1)


n2 – n = 2r, where r = q (2q + 1)


n2 – n is divisible by 2.


n 2 – n is divisible by 2 for every integer n, Hence proved


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