MCQ
$n^2 - 1$ is divisible by $8$, if $n$ is:
- AAn integer.
- BA natural number.
- ✓An odd integer.
- DAn even integer.
✓
Answer
Correct option: C.
An odd integer.
Let $a=n^2-1$
Here $n$ can be even 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: $n =$ 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 $8$ .
Here $n$ can be even 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: $n =$ 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 $8$ .
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
In $\Delta$PQR, $\angle$Q = 90°, PQ = QR = $5 \sqrt{2}$ PR = 10 then ∠P _______
Choose the correct alternative answer for each of the following question. What is the probability of the event that a number chosen from $1$ to $100$ is a prime number?
→It is given that $\triangle\text{ABC}\sim\triangle\text{PQR}$ and $\frac{\text{BC}}{\text{QR}}=\frac{2}{3}$ then $\frac{\text{ar}(\triangle\text{PQR})}{\text{ar}(\triangle\text{ABC})}=?$
→If AB : BC = 3 : 5, then how many equal parts seg AC divided to get point B _______ .
The distance of the point $(4, 7)$ from the $y-$axis is:
→$\sqrt2$ is:
→The zeros of the polynomial $\text{x}^2-\sqrt2\text{x}-12$ are:
→If the sum of the circumference of two circles with radii $r_1$ and $r_2$ is equal to the circumference of a circle of radius $r$, then:
→Choose the correct alternative answer for each of the following sub questions. Sum of first five multiples of $3$ is$. . .$
→Select the appropriate alternative.
ΔABC and ΔDEF are equilateral triangles, A(ΔABC) : A(ΔDEF) = 1 : 2. If AB = 4 then what is length of DE?
ΔABC and ΔDEF are equilateral triangles, A(ΔABC) : A(ΔDEF) = 1 : 2. If AB = 4 then what is length of DE?