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Question Number 190284 by cortano12 last updated on 31/Mar/23
find the remainder if 4^(2023)    divides by 7
$$\mathrm{find}\:\mathrm{the}\:\mathrm{remainder}\:\mathrm{if}\:\mathrm{4}^{\mathrm{2023}} \: \\ $$$$\mathrm{divides}\:\mathrm{by}\:\mathrm{7} \\ $$
Answered by alcohol last updated on 31/Mar/23
4^3  ≡ 1(mod 7)  ⇒ (4^3 )^(674)  ≡ 1(mod 7)  ⇒ 4^(2022)  ≡ 1(mod 7)  ⇒ 4^(2023)  ≡ 4(mod 7)  Remainder = 4
$$\mathrm{4}^{\mathrm{3}} \:\equiv\:\mathrm{1}\left({mod}\:\mathrm{7}\right) \\ $$$$\Rightarrow\:\left(\mathrm{4}^{\mathrm{3}} \right)^{\mathrm{674}} \:\equiv\:\mathrm{1}\left({mod}\:\mathrm{7}\right) \\ $$$$\Rightarrow\:\mathrm{4}^{\mathrm{2022}} \:\equiv\:\mathrm{1}\left({mod}\:\mathrm{7}\right) \\ $$$$\Rightarrow\:\mathrm{4}^{\mathrm{2023}} \:\equiv\:\mathrm{4}\left({mod}\:\mathrm{7}\right) \\ $$$${Remainder}\:=\:\mathrm{4} \\ $$
Answered by BaliramKumar last updated on 31/Mar/23
(4^(2023) /7) = (2^(2×2023) /7) = (2^(4046) /7)   = (2^(3×1348+2) /7) = ((8^(1348) ×2^2 )/7)  = ((1^(1348) ×4)/7)= ((1×4)/7) = (4/7) = 4 (remainder)
$$\frac{\mathrm{4}^{\mathrm{2023}} }{\mathrm{7}}\:=\:\frac{\mathrm{2}^{\mathrm{2}×\mathrm{2023}} }{\mathrm{7}}\:=\:\frac{\mathrm{2}^{\mathrm{4046}} }{\mathrm{7}}\: \\ $$$$=\:\frac{\mathrm{2}^{\mathrm{3}×\mathrm{1348}+\mathrm{2}} }{\mathrm{7}}\:=\:\frac{\mathrm{8}^{\mathrm{1348}} ×\mathrm{2}^{\mathrm{2}} }{\mathrm{7}} \\ $$$$=\:\frac{\mathrm{1}^{\mathrm{1348}} ×\mathrm{4}}{\mathrm{7}}=\:\frac{\mathrm{1}×\mathrm{4}}{\mathrm{7}}\:=\:\frac{\mathrm{4}}{\mathrm{7}}\:=\:\mathrm{4}\:\left({remainder}\right) \\ $$

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