## Task B-1.1.

Let $x$ be the greatest common divisor of the numbers $2^{2023}$ and $2^{2024}$, and $y$ their least common multiple. Calculate $3 y - x - 2 \cdot \frac{x^{2}}{y}$ and express the result as a power of 2.
## Solution.

Obviously, $x=2^{2023}$, and $y=2^{2024}$.

Thus, we obtain the following sequence of equalities:

$$
\begin{aligned}
3 y-x-2 \cdot \frac{x^{2}}{y} & =3 \cdot 2^{2024}-2^{2023}-2 \cdot \frac{\left(2^{2023}\right)^{2}}{2^{2024}} & & \\
& =6 \cdot 2^{2023}-2^{2023}-2 \cdot \frac{2^{4046}}{2^{2024}} & & 1 \text { point } \\
& =5 \cdot 2^{2023}-2 \cdot 2^{2022} & & 1 \text { point } \\
& =5 \cdot 2^{2023}-2^{2023} & & 1 \text { point } \\
& =4 \cdot 2^{2023}=2^{2025} & & 1 \text { point }
\end{aligned}
$$