Question

Prove that n square - n is divisible by 2 for every positive integer n.
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Answer 1

Answer: Case 1: Let n be an even positive integer.

When n = 2q  

In this case , 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 2: Let n be an odd positive integer.

When n = 2q + 1

In this case  

n2 -n = (2q + 1)2 - (2q + 1)= (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

Step-by-step explanation: Hence Proved