The remainder when $7^{2022}+3^{2022}$ is divided by 5 is :
Solution
$$
\begin{aligned}
& 7^{2022}+3^{2022} \\\\
& =\left(7^2\right)^{1011}+\left(3^2\right)^{1011} \\\\
&=(50-1)^{1011}+(10-1)^{1011} \\\\
&= (50^{1011}-1011.50^{1010}+\ldots-1) \\\\
& + (10^{1011}-1011.10^{1010}+\ldots . .-1) \\\\
&= 5 m-1+5 n-1=5(m+n)-2 \\\\
&= 5(m+n)-5+3=5(m+n-1)+3 \\\\
&= 5 k+3 \\\\
& \therefore \text { Remainder }=3
\end{aligned}
$$
About this question
Subject: Mathematics · Chapter: Binomial Theorem · Topic: Binomial Expansion
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