Question

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

Answer 1

n(n-1) is the product of two successive integers. it is always even number so it is divisible by 2

Answer 2

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

n^2  −n=4q^2 −2q

n^2  −n=2q(2q−1)

n^2  −n= 2r , where r=q(2q−1)

n^2  −n is divisible by 2 .

Case ii: Let n be an odd positive integer.

When n=2q+1

In this case  

n^2  −n= (2q+1)^2  −(2q+1)

n^2  −n=(2q+1)(2q+1−1)

n^2  −n =2q(2q+1)

n^2  −n =2r, where r=q(2q+1)

n^2 −n is divisible by 2.

∴ n^2 −n is divisible by 2 for every integer n