Task 3. Given the digits $0,1,3,4,5$. Using these digits, write down all the five-digit numbers with different digits that are divisible by 4 but not divisible by 5.
Solution. Let the form of the numbers be $\overline{a b c d e}$, where $a, b, c, d, e \in\{0,1,3,4,5\}$ and these digits are mutually distinct. Since the number $\overline{a b c d e}$ must be a five-digit number, it follows that $a \neq 0$. From $4 \mid \overline{a b c d e}$, it follows that $4 \mid \overline{d e}$, i.e., $\overline{d e} \in\{04,40\}$. From $5 \nmid \overline{a b c d e}$, it follows that $e \notin\{0,5\}$. Therefore, $\overline{d e}=04$, i.e., $d=0$ and $e=4$. The remaining digits are $1,3,5$ from which the three-digit beginning must be formed. All possible ways to do this are $\overline{a b c} \in\{135,153,315,351,513,531\}$.

Thus, the required five-digit numbers are 13504, 15304, 31504, 35104, $51304,53104$.