CBSE BoardEnglish MediumSTD 10MathsReal Numbers1 Mark
MCQ
If $m^2-1$ is divisible by $8,$ then $m$ is :
A
an even integer
B
a whole number
✓
an odd integer
D
a natural number
✓
Answer
Correct option: C.
an odd integer
Let $a=m^2-1$
Here $m$ can be ever or odd.
Case $I$ :
$m=$ Even i.e., $m=2 k$, 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 $\mathrm{k}=0, \mathrm{a}=4(0)^2-1=0-1=-1$, which is not divisible by $8$ , which is not.
Case $II$ :
$\mathrm{m}=$ Odd i.e., $\mathrm{m}=2 \mathrm{k}+1$, 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 the above two cases, if $m$ is odd, then $m^2-1$ is divisible by $8$ .
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.