The remainder when 428^2024 is divided by 21 is __________.

Question

The remainder when $428^{2024}$ is divided by 21 is \_\_\_\_\_\_\_\_\_\_.

Answer: 1

Accepted Answer

$\begin{gathered} 428=21	imes 20+8 \ \Rightarrow \, (428{)}^{2024}\equiv (20	imes 21+8{)}^{2024}\equiv {8}^{2024}(mod21) \ {8}^{2}=21	imes 3+1 \ {8}^{2024}=(21	imes 3+1{)}^{1012} \ \Rightarrow \, {8}^{2024}\equiv (21	imes 3+1{)}^{1012}(mod21) \ \equiv {1}^{2012}(mod21) \ {428}^{2024}\equiv 1(mod21) \end{gathered}$

Solutions · 1