Choose the correct answer from the given four options in the foll…

MCQ

Choose the correct answer from the given four options in the following questions:$n^2 – 1$ is divisible by 8, if n is:

A
An integer.
B
A natural number.
✓
An odd integer.
D
An even integer.

✓

Answer

Correct option: C.

An odd integer.

Let $a=n^2-1$
Here n can be ever or odd.
Case I:
$n =$ Even i.e., $n =2 k$. where k is an integer.
$\Rightarrow a=(2 k)^2-1$
$\Rightarrow a=4 k^2-1$
At $k=-1,=4(-1)^2-1=4-1=3$, which is not divisible by 8 .
At $k=0, a=4(0)^2-1=0-1=-1$, which is not divisible by 8 , which is not
Case II: $\square$
$n =$ Odd i.e., $n =2 k +1$, where k is an odd integer.
$\Rightarrow a=2 k+1$
$\Rightarrow a=(2 k+1) 2-1$
$\Rightarrow a=4 k^2+4 k+1-1$
$\Rightarrow a=4 k^2+4 k$
$\Rightarrow a=4 k(k+1)$
At $k=-1, a=4(-1)(-1+1)=0$ which is divisible by 8 .
At $k=0, a=4(0)(0+1)=4$ which is divisible by 8 .
At $k=1, a=4(1)(1+1)=8$ which is divisible by 8.
Hence, we can conclude from above two cases, if $n$ is odd, then $n^2-1$ is divisible by 8 .

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