The remainder when 32022 is divided by 5 is :

Question

Accepted Answer

${3^{2022}}$

$= {({3^2})^{1011}}$

$= {(9)^{1011}}$

$= {(10 - 1)^{1011}}$

$$ = {}^{1011}{C_0}{(10)^{1011}} + \,\,.....\,\, + \,\,{}^{1011}{C_{1010}}\,.\,{(10)^1} - {}^{1011}{C_{1011}}$$

$$ = 10\left[ {{}^{1011}{C_0}{{(10)}^{1010}} + \,\,......\,\, + \,\,{}^{1011}{C_{1010}}} \right] - 1$$

$$ = 10\,K - 1$$

[As $$10\left[ {{}^{1011}{C_0}\,.\,{{(10)}^{1010}} + \,\,......\,\, + \,\,{}^{1011}{C_{1010}}} \right]$$ is multiple of 10]

$= 10K + 5 - 5 - 1$

$= 10K - 5 + 5 - 1$

$= 5(2K - 1) + 4$

$$\therefore$$ Unit digit = 4 when divided by 5.

Solutions · 1