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Solve for $x $ : $$327x+208\equiv0 \mod 601$$

What I tried: I tried to find the modular inverse of $x$ first, so that I would be able to multiply the equation with it, but no luck so far.

The Demonix _ Hermit
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Kira02
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1 Answers1

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Use the Euclidean algorithm to determine the multiplicative inverse of $327$. We see $$ 601 = 1\cdot 601 + 0 \cdot 327 \\ 327 = 0\cdot 601 + 1 \cdot 327 \\ 274 = 1\cdot 601 + -1 \cdot 327 \\ 53 = -1\cdot 601 + 2 \cdot 327 \\ 9 = 6\cdot 601 + -11 \cdot 327 \\ -1 = -37 \cdot 601 + 68 \cdot 327 \\ 1 = 37 \cdot 601 + -68 \cdot 327. $$ Thus $327^{-1}\equiv -68 \equiv 533 \mod 601$. Now it is simple, rewrite the equation $$ 327 x \equiv - 208 \mod 601. $$ Multiply by the inverse of $327$ this results in $$ x \equiv -208\cdot 533 \equiv 321 \mod 601. $$

Math
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  • @OP That forward version of the Extended Euclidean Algorithm is explained at length here. There are over $150$ linked worked examples in its "Linked" questions list. – Bill Dubuque Jan 08 '20 at 20:40
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    you forgot an inverse symbol ... @Math $327\equiv -274\bmod 601$ –  Jan 08 '20 at 20:54
  • Thank you, I understand now. I tried the same thing and got stuck at 9 due to some flaws in my understanding of this algorithm but now I get it. Thanks again – Kira02 Jan 09 '20 at 05:48