Question

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

Solution

We know that any odd positive integer is of the form
49 +1 or, 4q +3 for some integer q.
So, we have the following cases :
Case I: When n=4g + 1
In this case, we have
n2 - 1 =(4q+ 1)2 - 1 = 16 q2 +8q+1-1
= 1692 +8q=8q (2q+1)
= n2-1 is divisible by 8 [: 89 (2q + 1) is divisible by 8]
Case II : When n=4q+3
In the case, we have
n2 - 1 =(49 +3)2 - 1 = 16q2 + 24q+9-1
= 16q2 +249 +8
= n2-1 = 8(2q2 +39 +1)=8 (2q + 1) (q+1)
= n2 - 1 is divisible by 8
: 8(2q + 1) (9+1) is divisible by 87
Hence, n2-1 is divisible by 8.