If n is an odd integer,then n^{2}-1 is divisi | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) Any odd integer $n$ can be expressed in the form $2k+1$ where $k$ is an integer.Alternatively,any odd integer $n$ can be represented as $n = 4k \p

Vedclass · Accepted Answer

$8$

Vedclass · Answer

(D) Any odd integer $n$ can be expressed in the form $2k+1$ where $k$ is an integer.Alternatively,any odd integer $n$ can be represented as $n = 4k \pm 1$ or simply by testing values.Let $n = 1$,then $n^{2}-1 = 1^{2}-1 = 0$ (divisible by $8$).Let $n = 3$,then $n^{2}-1 = 3^{2}-1 = 9-1 = 8$ (divisible by $8$).Let $n = 5$,then $n^{2}-1 = 5^{2}-1 = 25-1 = 24$ (divisible by $8$).General proof: Since $n$ is odd,$n = 2k+1$.$n^{2}-1 = (2k+1)^{2}-1 = 4k^{2}+4k+1-1 = 4k(k+1)$.Since $k(k+1)$ is the product of two consecutive integers,it is always even,i.e.,$k(k+1) = 2m$.Therefore,$n^{2}-1 = 4(2m) = 8m$.Thus,$n^{2}-1$ is always divisible by $8$.

If $n$ is an odd integer,then $n^{2}-1$ is divisible by...........

Similar Questions

The pair .......... is a like surds.

Prove that the number $\sqrt{3}+\sqrt{7}$ is irrational.

Why are the following numbers composite? Explain: $7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 + 5$

Find the largest number which divides $245$ and $1029$ leaving remainder $5$ in each case.

Find the $\text{l.c.m.}$ and $\text{g.c.d.}$ of the following by using the fundamental theorem of arithmetic: $84$,$90$,and $120$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module