CBSE Class 10 Answered

If 'n' is an odd positive integer, show that (n²-1) Is divisible by 8.
 
Any odd positive integer is in the form of 4p + 1 or 4p+ 3 for some positive integer p.
 
Let n = 4p+ 1,
 
(n² – 1) = (4p + 1)² – 1 = 16p² + 8p + 1 - 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.