Prove that n^2 - n is divisible by 2 for every positive integer n.
Answers
Given :
To prove that n² - n is divisible by 2 for every positive integer .
We know that all integers can be written in the form 2 q or 2 q + 1 .
q is some integer .
So , n = 2 q or n = 2 q + 1
When n = 2 q
n² - n = n ( n - 1 )
= 2 q ( 2 q - 1 )
Multiply the q :-
= 2 ( 2 q² - q )
==> ( n² - n ) / 2 = 2 q² - q
As 2 q² - q is an integer we have to say that n² - n is divisible by 2.
When n = 2 q + 1
n² - n = n ( n - 1 )
= ( 2 q + 1 )( 2 q + 1 - 1 )
= ( 2 q + 1 )( 2 q )
= 2 ( 2 q² + q )
==> ( n² - n ) / 2 = 2 q² + q
Again 2 q² + q is an integer , so n² - n must be divisible by 2
Hence n² - n is always divisible by 2 when n is a positive integer .
Hope it helps !
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