Wednesday, April 6, 2016

elementary number theory - Linear diophantine equation 100x23y=19




I need help with this equation: 100x23y=19. When I plug this into Wolfram|Alpha, one of the integer solutions is x=23n+12 where n is a subset of all the integers, but I can't seem to figure out how they got to that answer.


Answer



100x23y=19 if and only if 23y=100x+19, if and only if 100x+19 is divisible by 23. Using modular arithmetic, you have
\begin{align*} 100x + 19\equiv 0\pmod{23}&\Longleftrightarrow 100x\equiv -19\pmod{23}\\ &\Longleftrightarrow 8x \equiv 4\pmod{23}\\ &\Longleftrightarrow 2x\equiv 1\pmod{23}\\ &\Longleftrightarrow x\equiv 12\pmod{23}. \end{align*}
so x=12+23n for some integer n.



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