Question

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

Answer 1

Answer:

plzz. follow

Step-by-step explanation:

is of the form 4q+1 or. 4q + 3  for some integer q.

So, we have the following cases:

Case I   when n=4q+1

In this case, we have

n² - 1 = (4q + 1)² - 1 = 16q² + 8q + 1 – 1 = 16q² + 8q = 8q (2q + 1)

= n² - 1 is divisible by 8         [˙.˙ 8q(2q+1) is divisible by 8]

Case II  when n=4q+3

In this case, we have

n² -1 = (4q + 3)² - 1 = 16q² + 24q + 9 – 1 = 16q² + 24q + 8

= n² - 1 = 8(2q² + 3q + 1) = 8(2q + 1 )(q + 1)

=  n² - 1 is divisible by 8

Hence , n² - 1 is divisible by 8.

Answer 2

See the photo attached for solution