Let two consecutive positive integers are n - 1 and n

Now, product of two integers = n(n - 1) = n² - n

Case 1: when n is even

Let n = 2q

Now, n² - n = (2q)² - 2q

                 = 4q² - 2q

                 = 2q(2q - 1)

which is divisible by 2

Hence, the product n² - n is divisible by 2

Case 2: when n is odd

Let n = 2q + 1

Now, n² - n = (2q + 1)² - (2q + 1)

                 = 4q² + 4q + 1 - 2q - 1

                 = 4q² + 2q

                 = 2q(2q + 1)

which is divisible by 2

Hence, the product n² - n is divisible by 2

Hence, we can conclude that the product of two consecutive integers is divisible by 2