Question

2. Show that
square − 1 is divisible by 8, if is odd positive integer.​

Answer 1

Answer:

Any odd positive number is in the form of (4p+1)or(4p+3) for some integer P.

let

n=4p+3

n  

2

−1=(4p+1)  

2

−1=16p  

2

+8p+1−1=8p(2p+1)

⇒n  

2

−1isdivisibleby8

n  

2

−1=(4p+3)  

2

−1=16p  

2

+24p+9−1=16p  

2

+24p+8

=8(2p  

2

+3p+1)

⇒n  

2

−1isdivisibleby8

​  

 

Therefore, n  

2

−1 is divisible by 8 if n is an odd positive integer.