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Question Number 190284 by cortano12 last updated on 31/Mar/23

$$\mathrm{find}\mathrm{the}\mathrm{remainder}\mathrm{if}{4}^{2023}$$$$\mathrm{divides}\mathrm{by}7$$

Answered by alcohol last updated on 31/Mar/23

$$4^3\equiv 1\left(mod7\right)$$$$\Rightarrow {\left(4^3\right)}^{674}\equiv 1\left(mod7\right)$$$$\Rightarrow 4^{2022}\equiv 1\left(mod7\right)$$$$\Rightarrow 4^{2023}\equiv 4\left(mod7\right)$$$$Remainder=4$$

Answered by BaliramKumar last updated on 31/Mar/23

$$\frac{4^{2023}}{7}=\frac{2^{2\times 2023}}{7}=\frac{2^{4046}}{7}$$$$=\frac{2^{3\times 1348+2}}{7}=\frac{8^{1348}\times 2^2}{7}$$$$=\frac{1^{1348}\times 4}{7}=\frac{1\times 4}{7}=\frac{4}{7}=4\left(remainder\right)$$

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