The remainder when 7^2022+3^2022 is divided by 5 is:

Question

The remainder when $7^{2022}+3^{2022}$ is divided by 5 is:

Accepted Answer

$$
\begin{aligned}
& 7^{2022}+3^{2022} \\
& =\left(7^2\right)^{1011}+\left(3^2\right)^{1011} \\
&=(50-1)^{1011}+(10-1)^{1011} \\
&= (50^{1011}-1011.50^{1010}+\ldots-1) \\
& + (10^{1011}-1011.10^{1010}+\ldots . .-1) \\
&= 5 m-1+5 n-1=5(m+n)-2 \\
&= 5(m+n)-5+3=5(m+n-1)+3 \\
&= 5 k+3 \\
& \therefore \text { Remainder }=3
\end{aligned}
$$

Solutions · 1