Question

Prove that n2 – n is divisible by 2 for every positive integer n.

Answer 1

Hi Mate !!


Here's ur ans of above question !

Let n be any positive integer Which when divided by 2 gives q as quotient and r as remainder.

According to Euclid's division lemma

a = bq + r

Where , 0 ≤ r < b

So , n = 2q + r
r = 0 , 1

n = 2q

n = 2q + 1

__________

• Case - 1

n = 2q

n² - n = ( 2q )² - 2q

= 4q² - 2q

= 2 ( 2q² - q )

= 2m [ Where m = 2q² - q ]

Therefore It is divisible by 2


• Case - 2

n = 2q + 1

n² - n = ( 2q+ 1 )² - ( 2q+ 1 )

{ Using identity ( a + b )² = a² + b² + 2ab }

= 4q² + 1 + 4q - 2q - 1

= 4q² + 2q

= 2 ( 2q² + q )

= 2 m [ Where m = 2q² + q ]


Therefore It is divisible by both 2 !!

Hence proved !!