# 3. The numbers $2^{2021}$ and $5^{2021}$ are written one after another. How many digits are written in total?

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# Solution

Let the number $2^{2021}$ have $\mathrm{k}$ digits, and the number $5^{2021}$ have $\mathrm{m}$ digits, then the number of digits in the desired number is $\mathrm{k}+\mathrm{m}$. $10^{k-1}<2^{2021}<10^{k}, 10^{m-1}<5^{2021}<10^{m}$, therefore, $10^{k+m-2}<$ $10^{2021}<10^{m+k}$ and $\mathrm{k}+\mathrm{m}=2022$.

## Evaluation Criteria

The solution includes intermediate estimates that allow reaching the required estimate - from 13 points depending on the progress in the solution.

An incorrect answer due to a logical error - from 1-4 points depending on the progress in the solution.

An incorrect answer due to a computational error - 5 points.

A correct result obtained with justification - 7 points.