Confirming a number theoretical result

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Middle School Math Grades 5-8, Ages 10-13, MATHCOUNTS, AMC 8

Grades 5-8, Ages 10-13, MATHCOUNTS, AMC 8

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Middle School Math Grades 5-8, Ages 10-13, MATHCOUNTS, AMC 8

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Confirming a number theoretical result

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#1 h Yesterday at 5:10 AM • 1 Y

Y by aidan0626

Prove that $a \cdot c^{-1}+b \cdot d^{-1} = (ad+bc) \cdot (cd)^{-1} \pmod n$ where $\gcd(c,n) = \gcd(d,n) = 1$ .

This post has been edited 2 times. Last edited by OlympusHero, Yesterday at 5:10 AM

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#2 h Yesterday at 5:14 AM

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Note that $(cd)c^{-1}d^{-1}\equiv(c\cdot c^{-1})(d\cdot d^{-1})\equiv 1\pmod n$ , so $(cd)^{-1}\equiv c^{-1}d^{-1}\pmod n$ by definition, which basically proves the result.

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