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2-2024-2024-Remainder-




Question Number 208951 by hardmath last updated on 28/Jun/24
2^(2024)  : 2024 = ... (Remainder = ?)
$$\mathrm{2}^{\mathrm{2024}} \::\:\mathrm{2024}\:=\:…\:\left(\mathrm{Remainder}\:=\:?\right) \\ $$
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
thank you dear professor  answer: 2024?
$$\mathrm{thank}\:\mathrm{you}\:\mathrm{dear}\:\mathrm{professor} \\ $$$$\mathrm{answer}:\:\mathrm{2024}? \\ $$
Commented by A5T last updated on 28/Jun/24
1864
$$\mathrm{1864} \\ $$
Commented by hardmath last updated on 30/Jun/24
Dear professor, demon residue theorem?    Demon residue theorem
$$\mathrm{Dear}\:\mathrm{professor},\:\mathrm{demon}\:\mathrm{residue}\:\mathrm{theorem}? \\ $$$$ \\ $$Demon residue theorem
Commented by hardmath last updated on 30/Jun/24
dear professor
$$\mathrm{dear}\:\mathrm{professor} \\ $$
Commented by A5T last updated on 01/Jul/24
Chinese Remainder Theorem is the method I  know this is called.
$${Chinese}\:{Remainder}\:{Theorem}\:{is}\:{the}\:{method}\:{I} \\ $$$${know}\:{this}\:{is}\:{called}. \\ $$

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