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If n is an odd integer, then show that $n 2 − 1$ is divisible by

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answer is C.

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Detailed Solution

It is given that

n

is an odd integer.
According to Euclid’s division lemma,
Given positive integers say a and b, there exist unique integers q and r which satisfies

a = bq + r

, where

0 \leq r < m

.
Since

n

is an odd integer.

∴ n = 1, 3, 5, ...

.
So, for n=1,

n^{2} - 1 = 1 - 1 = 0

Here,

n 2 − 1

is divisible by 8.
For n=3,

n^{2} - 1 = 9 - 1 = 8

Here,

n 2 − 1

is divisible by 8.
For n=5,

n^{2} - 1 = 25 - 1 = 24

Here, 24 is divisible by 8.
So,

n 2 − 1

is divisible by 8.
Hence, it is proved that

n

is an odd integer, then

n 2 − 1

is divisible by 8.
Therefore, the correct option is 3.

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