Question

Prove that any positive integer n cube - n is divisible by 2

Answer 1

Answer:

Here is the answer to your question.

 Any positive integer is of the form 2q or 2q + 1, where q is some integer

When n = 2q, 

n²+n=(2q)²+2q

      =4q²+2q

      = 2q(2q+1)   

which is divisible by 2

 

when n=2q+1

n²-n=(2q+1)²+(2q+1)

      = 4q²+4q+1+2q+1

      =4q²+6q +2

     =  2(2q²+3q+1)

which is divisible by 2

hence n²+n is divisible by 2 for every positive integer n