I need help with this equation: 100x−23y=−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
100x−23y=−19 if and only if 23y=100x+19, if and only if 100x+19 is divisible by 23. Using modular arithmetic, you have
100x+19≡0(mod23)⟺100x≡−19(mod23)⟺8x≡4(mod23)⟺2x≡1(mod23)⟺x≡12(mod23).
so x=12+23n for some integer n.
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