Question

Let $p\left(n\right)=4\cdot 6^n+5^{n+1}$. When $P\left(n\right)$ is divided by $20$, leaves the remainder

• A. $8$
• B. $9$
• C. $0$
• D. none of these

Answer 1

The correct option is B $9$

$p\left(n\right)=4.6^n+5^{n+1}=4{(1+5)}^n+5.{(1+4)}^n$
$\Rightarrow p\left(n\right)=4(1+n_{C_1}5+n_{C_2}{5}^2+...+n_{C_n}{5}^n)+5(1+n_{C_1}4+n_{C_2}{4}^2+...+n_{C_n}{4}^n)$
$\Rightarrow p\left(n\right)=9+20[(n_{C_1}+n_{C_2}5+...+n_{C_n}{5}^{n-1})+(n_{C_1}+n_{C_2}4+...+n_{C_n}{4}^{n-1})]$
Therefore, remainder is $9$ when divisible by $20$.
Ans: B