2. It is known that a two-digit number, when divided by 3, gives a remainder of 1, and when divided by 5, gives a remainder of 3. Find all such numbers.
Answer: $\{13 ; 23 ; 43 ; 58 ; 73 ; 88\}$.

## Solution:

Let the required number be $A$, then we have $A=3 m+1=5 n+3$, from which we obtain the linear Diophantine equation $3 m-5 n=2$. Easily selecting a particular solution $m=n=-1$, we get the general solution in the form

$$
\left\{\begin{array}{l}
m=-1+5 t, \\
n=-1+3 t,
\end{array} \quad t \in Z . \quad \Rightarrow \quad A=15 t-2, t \in Z\right.
$$

Considering that $A$ is a two-digit number, we get $t=\overline{1,6}$. By trying $t=\overline{1,6}$, we obtain all the required numbers $\{13 ; 23 ; 43 ; 58 ; 73 ; 88\}$.