The remainder when 32022 is divided by 5 is :
Solution
<p>${3^{2022}}$</p>
<p>$= {({3^2})^{1011}}$</p>
<p>$= {(9)^{1011}}$</p>
<p>$= {(10 - 1)^{1011}}$</p>
<p>$$ = {}^{1011}{C_0}{(10)^{1011}} + \,\,.....\,\, + \,\,{}^{1011}{C_{1010}}\,.\,{(10)^1} - {}^{1011}{C_{1011}}$$</p>
<p>$$ = 10\left[ {{}^{1011}{C_0}{{(10)}^{1010}} + \,\,......\,\, + \,\,{}^{1011}{C_{1010}}} \right] - 1$$</p>
<p>$= 10\,K - 1$</p>
<p>[As $$10\left[ {{}^{1011}{C_0}\,.\,{{(10)}^{1010}} + \,\,......\,\, + \,\,{}^{1011}{C_{1010}}} \right]$$ is multiple of 10]</p>
<p>$= 10K + 5 - 5 - 1$</p>
<p>$= 10K - 5 + 5 - 1$</p>
<p>$= 5(2K - 1) + 4$</p>
<p>$\therefore$ Unit digit = 4 when divided by 5.</p>
About this question
Subject: Mathematics · Chapter: Binomial Theorem · Topic: Binomial Expansion
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