Question

Show that n²-1 is divisible by 8, if n is an odd positive integer.

Answer 1

hello,

  Any odd positive integer is in the form of 4p + 1 or 4p+ 3 for some integer p.  
 Let n = 4p+ 1,  (n² – 1)
= (4p + 1)² – 1
= 16p² + 8p + 1
= 16p² + 8p
= 8p (2p + 1) 
⇒ (n² – 1) is divisible by 8.   (n² – 1)
= (4p + 3)² – 1
= 16p² + 24p + 9 – 1
= 16p²+ 24p + 8
= 8(2p² + 3p + 1) 
⇒ n²– 1 is divisible by 8.  Therefore, n²– 1 is divisible by 8 if n is an odd positive integer.

Answer 2

n is odd positive integer imply it is of the form 2m+1. where m is positive integer.

proceed further, if it's possible