Question

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

Answer 1

Answer:

Let n be a positive interger and b = 2

By euclid division lemma:

n = 2q + r r = 0,2

Now, let n = 2q + 0

then, n^2 - n

= (2q)^2 - n

= 4q^2 - 2q

= 2( 2q^2 - q ) (2q^2 - q = m)

= 2m _______________________1

Let n = 2q + 1

then,

n^2 - n

= (2q + 1) ^ 2 - 2q + 1

= 4q^2 + 1 +2q - 2q + 1

= 4q^2 + 2

= 2(2q^2 + 1) (2q^2 + 1 = m)

= 2m ___________________________2

From 1st and 2nd since the value of n^2 - n is the multiple of 2 so it is divisible by 2.