Question

The remainder when the determinant
$\left|\begin{array}{ccc}2014^{2014}& 2015^{2015}& 2016^{2016}\\ 2017^{2017}& 2018^{2018}& 2019^{2019}\\ 2020^{2020}& 2021^{2021}& 2022^{2022}\end{array}\right|$
is divided by $5$ is

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

Answer: (D)

Let $D=\left|\begin{array}{ccc}2014^{2014}& 2015^{2015}& 2016^{2016}\\ 2017^{2017}& 2018^{2018}& 2019^{2019}\\ 2020^{2020}& 2021^{2021}& 2022^{2022}\end{array}\right|$
$D=\left|\begin{array}{ccc}{\left(2015-1\right)}^{2014}& {\left(2015\right)}^{2015}& {\left(2015+1\right)}^{2016}\\ {\left(2015+2\right)}^{2017}& {\left(2020-2\right)}^{2018}& {\left(2020-1\right)}^{2019}\\ {\left(2020\right)}^{2020}& {\left(2020+1\right)}^{2021}& {\left(2020+2\right)}^{2022}\end{array}\right|$
Remainder when divided by $5$, is
$D=\left|\begin{array}{ccc}1& 0& 1\\ 2^{2017}& 2^{2018}& -1\\ 0& 1& 2^{2022}\end{array}\right|$
$=1\left(2^{4040}+1\right)+2^{2017}$
$={\left(5-1\right)}^{2020}+1+2{\left(5-1\right)}^{1008}$
$=1+1+2=4$