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Solve-for-x-x-1-mod-3-x-4-mod-5-x-6-mod-7-




Question Number 115674 by naka3546 last updated on 27/Sep/20
Solve  for  x        x  ≡  1  mod  3        x  ≡  4  mod  5        x  ≡  6  mod  7
$${Solve}\:\:{for}\:\:{x} \\ $$$$\:\:\:\:\:\:{x}\:\:\equiv\:\:\mathrm{1}\:\:{mod}\:\:\mathrm{3} \\ $$$$\:\:\:\:\:\:{x}\:\:\equiv\:\:\mathrm{4}\:\:{mod}\:\:\mathrm{5} \\ $$$$\:\:\:\:\:\:{x}\:\:\equiv\:\:\mathrm{6}\:\:{mod}\:\:\mathrm{7} \\ $$
Answered by floor(10²Eta[1]) last updated on 27/Sep/20
x≡1(mod3)⇒x=3a+1  3a+1≡4(mod5)⇒3a≡3(mod5)⇒a≡1(mod5)  a=5b+1⇒x=3(5b+1)+1=15b+4  15b+4≡6(mod7)⇒15b≡b≡2(mod7)  b=7c+2⇒x=15(7c+2)+4  ⇒x=105c+34 ∀ c∈Z
$$\mathrm{x}\equiv\mathrm{1}\left(\mathrm{mod3}\right)\Rightarrow\mathrm{x}=\mathrm{3a}+\mathrm{1} \\ $$$$\mathrm{3a}+\mathrm{1}\equiv\mathrm{4}\left(\mathrm{mod5}\right)\Rightarrow\mathrm{3a}\equiv\mathrm{3}\left(\mathrm{mod5}\right)\Rightarrow\mathrm{a}\equiv\mathrm{1}\left(\mathrm{mod5}\right) \\ $$$$\mathrm{a}=\mathrm{5b}+\mathrm{1}\Rightarrow\mathrm{x}=\mathrm{3}\left(\mathrm{5b}+\mathrm{1}\right)+\mathrm{1}=\mathrm{15b}+\mathrm{4} \\ $$$$\mathrm{15b}+\mathrm{4}\equiv\mathrm{6}\left(\mathrm{mod7}\right)\Rightarrow\mathrm{15b}\equiv\mathrm{b}\equiv\mathrm{2}\left(\mathrm{mod7}\right) \\ $$$$\mathrm{b}=\mathrm{7c}+\mathrm{2}\Rightarrow\mathrm{x}=\mathrm{15}\left(\mathrm{7c}+\mathrm{2}\right)+\mathrm{4} \\ $$$$\Rightarrow\mathrm{x}=\mathrm{105c}+\mathrm{34}\:\forall\:\mathrm{c}\in\mathbb{Z} \\ $$

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