in-solving-the-linear-congruence-ax-b-mod-n-n-ax-b-ax-b-kn-ax-kn-b-solving-the-linear-diophantine-equation-ax-kn-b-what-are-the-general-solution-to-the-equation-ax-kn-b-

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Question Number 93184 by Rio Michael last updated on 11/May/20

in solving the linear congruence ax ≡ b (mod n) ⇒ n∣(ax − b) ⇒ ax −b = kn ⇔ ax −kn = b ⇒ solving the linear diophantine equation ax −kn = b what are the general solution to the equation ax−kn = b

$$\mathrm{in}\:\mathrm{solving}\:\mathrm{the}\:\mathrm{linear}\:\mathrm{congruence} \\ $$$${ax}\:\equiv\:{b}\:\left(\mathrm{mod}\:{n}\right)\:\Rightarrow\:{n}\mid\left({ax}\:−\:{b}\right)\:\Rightarrow\:{ax}\:−{b}\:=\:{kn}\:\Leftrightarrow\:{ax}\:−{kn}\:=\:{b} \\ $$$$\Rightarrow\:\mathrm{solving}\:\mathrm{the}\:\mathrm{linear}\:\mathrm{diophantine}\:\mathrm{equation}\:{ax}\:−{kn}\:=\:{b} \\ $$$$\:\mathrm{what}\:\mathrm{are}\:\mathrm{the}\:\mathrm{general}\:\mathrm{solution}\:\mathrm{to}\:\mathrm{the}\:\mathrm{equation} \\ $$$$\:{ax}−{kn}\:=\:{b} \\ $$$$\: \\ $$$$ \\ $$

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