Question

show that n square minus n is divisible by 2 for every positive integer n​

Answer 1

Casei: 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

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 Case ii: 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