Difference between revisions of "2023 SSMO Relay Round 1 Problems/Problem 1"

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Notice that over mod <math>2023</math>, we have <math>2022^{2021^{2020^{\dots}}}\equiv(-1)^{2021^{2020^{\dots}}}</math>. Since the power is odd, we conclude that the remainder must be <math>-1\equiv\boxed{2022}</math>.
 
Notice that over mod <math>2023</math>, we have <math>2022^{2021^{2020^{\dots}}}\equiv(-1)^{2021^{2020^{\dots}}}</math>. Since the power is odd, we conclude that the remainder must be <math>-1\equiv\boxed{2022}</math>.
  
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Determine the value of <math>1+\frac{1}{2+\frac{1}{1+\frac{1}{2+\frac{1}{1+\cdots}}}}</math>

Revision as of 20:30, 19 March 2025

Problem

Compute the remainder when $2022^{2021^{2020^{\dots}}}$ is divided by $2023$.

Solution

Notice that over mod $2023$, we have $2022^{2021^{2020^{\dots}}}\equiv(-1)^{2021^{2020^{\dots}}}$. Since the power is odd, we conclude that the remainder must be $-1\equiv\boxed{2022}$.

Determine the value of $1+\frac{1}{2+\frac{1}{1+\frac{1}{2+\frac{1}{1+\cdots}}}}$