Prove that the product of two consecutive numbers is divisible by 2

1.	Prove that the product of two consecutive numbers is divisible by 2
Answer» Let n-1 and n be consecutive positive integers,Let P be their productThen {tex}\\style{font-family:Arial}{\\style{font-size:12px}{\\mathrm n(\\mathrm n-1)=\\mathrm n^2-1\\;.................(1)}}{/tex}We know that any positive integers is of the form 2q or 2q + 1, where q is a positive integerCase I: When n = 2q, thenP=n2\xa0- n = (2q)2\xa0- 2q = 4q2\xa0- 2q = 2q(2q - 1)-----(2)Case II: When n = 2q + 1, thenP=n2\xa0- n = (2q + 1)2\xa0- (2q + 1)= 4q2\xa0+ 4q + 1 - 2q - 1= 4q2\xa0+ 2qP = 2q(2q + 1)..........(3)From (2) and (3) we conclude that the product of n-1 and n is divisible by 2

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