2-2024-2024-Remainder-

Question Number 208951 by hardmath last updated on 28/Jun/24

$$\mathrm{2}^{\mathrm{2024}} \::\:\mathrm{2024}\:=\:…\:\left(\mathrm{Remainder}\:=\:?\right) \\ $$

Answered by A5T last updated on 28/Jun/24

(2^(11) )^(184) ≡^(2024) 24^(184) 2024=8×11×23 24^(184) ≡2^(184) ≡32^(36) 2^4 ≡5(mod 11) 24^(184) ≡0(mod 8) 24^(184) ≡1(mod 23) 23k+1≡0(mod 8)⇒k=8q+1 23(8q+1)+1=23×8q+24≡5(mod 11) 8q≡3(mod 11)⇒q≡10(mod 11)⇒q=11c+10 2^(2024) =2024c+23×8×10+24=2024c+1864 ⇒2^(2024) ≡1864(mod 2024)

$$\left(\mathrm{2}^{\mathrm{11}} \right)^{\mathrm{184}} \:\:\overset{\mathrm{2024}} {\equiv}\:\mathrm{24}^{\mathrm{184}} \\ $$$$\mathrm{2024}=\mathrm{8}×\mathrm{11}×\mathrm{23} \\ $$$$\mathrm{24}^{\mathrm{184}} \equiv\mathrm{2}^{\mathrm{184}} \equiv\mathrm{32}^{\mathrm{36}} \mathrm{2}^{\mathrm{4}} \equiv\mathrm{5}\left({mod}\:\mathrm{11}\right) \\ $$$$\mathrm{24}^{\mathrm{184}} \equiv\mathrm{0}\left({mod}\:\mathrm{8}\right) \\ $$$$\mathrm{24}^{\mathrm{184}} \equiv\mathrm{1}\left({mod}\:\mathrm{23}\right) \\ $$$$\mathrm{23}{k}+\mathrm{1}\equiv\mathrm{0}\left({mod}\:\mathrm{8}\right)\Rightarrow{k}=\mathrm{8}{q}+\mathrm{1} \\ $$$$\mathrm{23}\left(\mathrm{8}{q}+\mathrm{1}\right)+\mathrm{1}=\mathrm{23}×\mathrm{8}{q}+\mathrm{24}\equiv\mathrm{5}\left({mod}\:\:\mathrm{11}\right) \\ $$$$\mathrm{8}{q}\equiv\mathrm{3}\left({mod}\:\mathrm{11}\right)\Rightarrow{q}\equiv\mathrm{10}\left({mod}\:\mathrm{11}\right)\Rightarrow{q}=\mathrm{11}{c}+\mathrm{10} \\ $$$$\mathrm{2}^{\mathrm{2024}} =\mathrm{2024}{c}+\mathrm{23}×\mathrm{8}×\mathrm{10}+\mathrm{24}=\mathrm{2024}{c}+\mathrm{1864} \\ $$$$\Rightarrow\mathrm{2}^{\mathrm{2024}} \equiv\mathrm{1864}\left({mod}\:\mathrm{2024}\right) \\ $$

Commented by hardmath last updated on 28/Jun/24