Question

Every even power of every odd number greater than $1$ when divided by $8$ leaves the remainder

• A. $7$
• B. $3$
• C. $5$
• D. $1$

Answer 1

The correct option is D $1$

Consider such a number given by :
$(2k+1{)}^{2n}$
$=(4k^2+4k+1{)}^n$
$=\left(4\right(k^2+k)+1{)}^n$
$k^2+k=k(k+1)$ is an even number$=2p($say$)$
$\Rightarrow (2k+1{)}^{2n}=(8p+1{)}^n$
Expanding by binomial theorem, all the terms except the last term $1$ is divisible by $8$.
$\therefore$ Remainder will be $1$