Question

Prove that the product of two consecutive positive integers is divisible by 2.

Answer 1

Let the two consecutive positive integers be x and (x+1)
Product of two consecutive positive integers = x(x+1)
$=x^2+x$
Case (i) : x is even number
Let x = 2k
$\Rightarrow x^2+x=(2k{)}^2+2k$
$4k^2+2k$
$=2k(2k+1)$
Hence the product is divisible by 2

Case (ii): x is odd number
Let x= 2k +1
$\Rightarrow x^2+x=(2k+1{)}^2+(2k+1)$
$=4k^2+6k+2$
$=2(2k^2+3k+1)$
Clearly the product is divisible by 2
From the both the cases we can conclude that the product of two consecutive integers is divisible by 2.