## Task B-1.1.

Determine the natural number $n$ such that the number $\left(4^{n-1}:\left(2 \cdot 8^{-n-1}\right)\right) \cdot\left(25^{n}: 5^{2 n-1}\right)^{5 n}-1$ has 10120 digits.
## Solution.

By simplifying the given expression, we get:

$\left(4^{n-1}:\left(2 \cdot 8^{-n-1}\right)\right) \cdot\left(25^{n}: 5^{2 n-1}\right)^{5 n}-1=$

$\left(2^{2 n-2}:\left(2^{-3 n-2}\right)\right) \cdot\left(5^{2 n}: 5^{2 n-1}\right)^{5 n}-1=$

$\left(2^{5 n}\right) \cdot\left(5^{5 n}\right)-1=$

$10^{5 n}-1$

Since $10^{5 n}=1 \underbrace{000 \ldots 0}_{5 n \text{ zeros }}$, it follows that $10^{5 n}-1=\underbrace{999 \ldots 9}_{5 n \text{ nines }}$.

Therefore, the number $\left(4^{n-1}:\left(2 \cdot 8^{-n-1}\right)\right) \cdot\left(25^{n}: 5^{2 n-1}\right)^{5 n}-1$ has $5 n$ digits.

According to the condition of the problem, $5 n=10120$, so we conclude that $n=2024$.