## Task A-3.7. (10 points)

Find all two-digit natural numbers $a$ for which the equation

$$
2^{x+y}=2^{x}+2^{y}+a
$$

has a solution $(x, y)$ in natural numbers.
## First Solution.

We transform the original equation:

$$
\left(2^{x}-1\right)\left(2^{y}-1\right)=a+1
$$

For the equation to have solutions, it is necessary and sufficient that $a+1$ is a product of factors of the form $2^{x}-1$, where $x$ is some natural number.

Since $a+1$ must be a two-digit number, these factors can only be $1, 3, 7, 15, 31,$ and $63$.

(3 points)

The possibilities for $a+1$ are as follows:

$$
15=1 \cdot 15, \quad 21=3 \cdot 7, \quad 31=1 \cdot 31, \quad 45=3 \cdot 15, \quad 49=7 \cdot 7, \quad 63=1 \cdot 63, \quad 93=3 \cdot 31
$$

Thus, the possible values for $a$ are:

$$
14, 20, 30, 44, 48, 62, 92
$$