Question

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

Answer 1

Answer:

Step-by-step explanation:

Case i: Let n be an even positive integer.

When n=2q

In this case , we have  

$n^{2}-n\\=(2q)^{2}-2q \\=4q^{2} -2q\\=2q(2q-1)\\=2r , r=q(2q-1)$ for any integer

it  is  multiple of 2

therefore it is divisible by 2

Case ii: Let n be an odd positive integer.

When n=2q+1

In this case

$=(2q+1)^{2} - (2q+1)\\=(2q+1)(2q+1-1)=2q(2q+1)\\=2r ,where , r=2q(2q+1)$

for any integer

it  is  multiple of 2

therefore it is divisible by 2