Question

Find the remainder when $6^n-5n$ is divided by $25$?

• A. $1$
• B. $24$
• C. $5$
• D. None of these

Answer 1

The correct option is A $1$

$6^n-5n=(1+5{)}^n-5n$

${}^n{C}_0+{}^n{C}_1\left(5\right)+\dots +{}^n{C}_n(5{)}^n-5n$

$1+\left[{}^n{C}_2\right(5{)}^2+{}^n\left({}_3\right(5{)}^3+\dots +{}^n{C}_n\left(5{)}^n\right]$

$1+25\left[{}^n{C}_2\right(5)+{}^n{C}_3(5{)}^2+\dots +{}^n\left({}_n\right(5{)}^{n-1}]$

$\text{so remainder is}1$