2023 SSMO Relay Round 1 Problems/Problem 1

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 $1+\frac{1}{2+\frac{1}{1+\frac{1}{2+\frac{1}{1+\cdots}}}}$

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2023 SSMO Relay Round 1 Problems/Problem 1

Problem

Solution