Question

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

Answer 1

Step-by-step explanation:

1x2=2

2\2=1

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Answer 2

To prove:-

• That the product of two consecutive integer is divisible by 2.

Proof:-

• Let n - 1 and n be two consecutive positive integer.

Then there product is (n - 1) = n² - n

We know that every positive integer is of the form 2q or 2q + 1 for some integer q.

So, Let n = 2q

So, n² - n = (2q)² - (2q)

=> n² - n = (2q)² - (2q)

=> n² - n = 4q² - 2q

=> n² - n = 2q(2q - 1)

=> n² - n = 2r [where r = q(2q - 1)]

=> n² - n is even and divisible by 2

Let n = 2q + 1

So, n² - n = (2q + 1)(2q + 1) - 1

=> n² - n = (2q + 1)(2q)

=> n² - n = 2r(r = q(2q + 1))

=> n² - n is even and divisible by 2

Hence, it is proved that the product of two consecutive integer is divisible by 2.