2. It is known that a two-digit number, when divided by 4, gives a remainder of 3, and when divided by 3, gives a remainder of 2. Find all such numbers.
Answer: $\{11 ; 23 ; 35 ; 47 ; 59 ; 71 ; 83 ; 95\}$.

## Solution:

Let the desired number be $A$, then we have $A=4 m+3=3 n+2$, from which we obtain the linear Diophantine equation $3 n-4 m=1$. Easily selecting a particular solution $m=n=-1$, we get the general solution in the form

$$
\left\{\begin{array}{l}
m=-1+3 t, \\
n=-1+4 t,
\end{array} \quad t \in Z . \quad \Rightarrow \quad A=12 t-1, t \in Z\right.
$$

Considering that $A$ is a two-digit number, we get $t=\overline{1,8}$. By trying $t=\overline{1,8}$, we obtain all the desired numbers $\{11 ; 23 ; 35 ; 47 ; 59 ; 71 ; 83 ; 95\}$.