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I am trying to solve:

$x = 1(\mod13), x = 2(\mod31)$. Using the chinese remainder theorem, I got $x = 31*8 + 13 *y$, where I don't know what $y$ is, since I don't know how to solve $x = 13(\mod31)$. Please help!

Alireza Firouzja
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  • See the first dupe link for how the CRT formula works, and the other links for various ways to compute the modular inverses in the formula. – Bill Dubuque Apr 19 '20 at 03:44

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According to the schema for the explicit construction of $x$, $y$ should be $2\cdot13^{-1}\bmod31$. We can find the inverse of $13$ using the extended Euclidean algorithm, and it works out to be $12$, so $y=24$.

Parcly Taxel
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