## Task 3 - 050833

Provide all quadruples $\left(z_{1}, z_{2}, z_{3}, z_{4}\right)$ of two-digit numbers $z_{1}, z_{2}, z_{3}, z_{4}$ that have the following properties. For each quadruple:

(1) $z_{1} \cdot z_{2}=z_{3} \cdot z_{4}$,

(2) $z_{3}$ is obtained by reading $z_{1}$ backwards,

(3) $z_{4}$ is obtained by reading $z_{2}$ backwards, (Example $24 \cdot 63=42 \cdot 36$ )

(4) Among the four digits of $z_{1}$ and $z_{2}$, there are no two that are the same,

(5) $z_{1}$ is the smallest of the four numbers.
If the digits of $z_{1}$ are denoted by $a$ and $b$, and those of $z_{2}$ by $c$ and $d$ (in this order), then according to the problem, $(10 a+b)(10 c+d)=(a+10 b)(c+10 d)$, which simplifies to $99 a c=99 b d$, and thus $a c=b d$.

Given $1 \leq a, b, e, d \leq 9$ (integers) and due to (4) and (5), the following 10 number quadruples that satisfy (1) are obtained:

| $(12$, | 63, | 21, | $36)$ | $(14$, | 82, | 41, | $28)$ |
| :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- |
| $(12$, | 84, | 21, | $48)$ | $(23$, | 64, | 32, | $46)$ |
| $(24$, | 63, | 42, | $36)$ | $(23$, | 96, | 32, | $69)$ |
| $(13$, | 62, | 31, | $26)$ | $(34$, | 86, | 43, | $68)$ |
| $(26$, | 93, | 62, | $39)$ | $(36$, | 84, | 63, | $48)$. |

Conversely, it is recognized that each of these quadruples satisfies the given conditions.