Question

If n > 1 and (n2 -1) is divisible by 8, what is the least value of n?

Answer 1

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

 

Let n=4p+1,

 

(n  

2

–1)=(4p+1)  

2

–1=16p  

2

+8p+1=16p  

2

+8p=8p(2p+1)

⇒(n  

2

–1) is divisible by 8.

 

(n  

2

–1)=(4p+3)  

2

–1=16p  

2

+24p+9–1=16p  

2

+24p+8=8(2p  

2

+3p+1)

⇒n  

2

–1 is divisible by 8.

 

Therefore, n  

2

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