## Task A-1.1.

Determine all pairs of non-negative integers $(a, b)$ that satisfy the equation:

$$
2^{a} \cdot 3^{b}-3^{b+1}+2^{a}=13
$$
## First solution.

We can write the equation in the form $\left(2^{a}-3\right) \cdot 3^{b}=13-2^{a}$.

Since $3^{b}>0$, the numbers $2^{a}-3$ and $13-2^{a}$ must have the same sign, so $3<2^{a}<13$.

This is satisfied only by the numbers $a=2$ and $a=3$.

For $a=2$, we have $1 \cdot 3^{b}=9$, so $b=2$.

For $a=3$, we have $5 \cdot 3^{b}=5$, so $b=0$.

Therefore, the only solutions are the pairs $(2,2)$ and $(3,0)$.