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Prove that the product of two consecutive positive integer is divisible by2 |
| Answer» Let the two consecutive positive integers be x and x + 1Product of two consecutive positive integers = x (x + 1)= x2 + xCase (i): x is even numberLet x = 2k⇒x2 + x = (2k)2 + 2k = 4k2 + 2k= 2k ( 2k + 1) Hence the product is divisible by 2Case (ii): x is odd numberLet x = 2k + 1⇒ x2 + x = (2k + 1)2 + (2k + 1) = 4k2 + 4k + 1 + 2k + 1= 4k2 + 6k + 2\xa0= 2 (2k2 + 3k + 1) Clearly the product is divisible by 2From the both the cases we can conclude that the product of two consecutive positive integers is divisible by 2. | |