CBSE BoardEnglish MediumSTD 10MathsReal Numbers1 Mark
MCQ
$n^2-1$ 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
(C) an odd integer If $n$ is an even integer, then so is $n^2$ and hence $n^2-1$ is an odd integer. Consequently, $n^2-1$ cannot be divisible by 8 , if $n$ is an even integer. So, let $n$ be an odd integer. Then, $n=2 m+1$ for some integer $m$. $ \therefore \quad n^2-1=(2 m+1)^2-1=4 m^2+4 m=4 m(m+1) $ For any natural number $m, m(m+1)$ is an even natural number. Let $m=2 k$, where $k$ is a natural number. $\therefore \quad n^2-1=8 k(2 k+1)$, which is divisible by 8 . Hence, $n$ is an odd integer.
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