If n is an odd integer then ` n ^(2) -1` is divisible by

Options

A. `8`
B. `7`
C. `3`
D. `9`

Answer

A

Explanation

We know that every odd integer is of the form 4q +1 or 4q +3 for some integer q.
Case I when n= 4q +1
Then `n^(2) -1 = ( 4q +1)^(2) -1`
Then `n ^(2) -1 = ( 4q +1)^(2) -1`
` 16q^(2) + 8q +1 -1`
` = 16q ^(2) + 8q`
` 8( q + 2q^(2))` which is divisible by 8.
Case II when n= 4q +3
then , ` n^(2) -1 ( 4q +3) ^(2) -1`
=` 16q^(2) +24q +9-1`
` 16q^(2) + 24q +8`
` = 8 ( 2q^(2) + 3q +1)` , whihc is divisible by 8.
Hence, if n is an odd integer then ` n^(2) -1` is divisble by 8.