Question

show that {n}^{2} - 1 is divisible by 8, if n is an odd positive integer.

Answer 1

Let n be 5(because 5 is an odd positive integer)
A/q
n²-1=5²-1=24
Now we know that 24 is a multiple of 8.
Hope it helps.

Answer 2

hey sister !!!


here is your answer !!

we know that any positive integer is in the form of 4q + 1 or 4q+3 .

so , we have two cases :-

Case 1 :-

n = 4q+ 1

according to this case .

n² -1 = ( 4q+ 1)² -1

=> n²-1 = 16 q² + 8q +1 -1

=>. 8q (2q+1) .


{ 8q (2q+1) is divisible by 8 }


hence n² - 1 is divisible by 8


now , we have case 2


when : 4 q+ n


n² - 1 = (4q +3 ) ² -1

=> 16 q² + 24 q +8


hence , n² - 1 = 8 ( 2q ² + 3q +1 )


clerly


n² -1 is divisible by 8 because. 8 ( 2q² + 3q +1) is divisible by 8



hence , n² -1 is divisible by 8 in both cases !!! =====================================

hence , proved !!


hope it helps you sister !!!

thanks !!!