1.

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

Answer»

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.


Discussion

No Comment Found

Related InterviewSolutions