Task 1. Find natural numbers $a$ and $b$, given that the number $\overline{a b b a}$ is a perfect cube.
Task 1. Find natural numbers $a$ and $b$, if it is known that the number $\overline{a b b a}$ is a perfect cube.

| Solution | Criteria |
| :---: | :---: |
| Note that the number $\overline{a b b a}$ is divisible by 11 (by the divisibility rule for 11). Since $22^{3}=10648$ is a five-digit number, $\overline{a b b a}=11^{3}=1331$. Answer: $\mathrm{a}=1, \mathrm{~b}=3$ | 5 points for the correct answer. 1 point if it is proven that $\overline{a b b a}$ is divisible by 11. 2 points if it is proven that there are no other solutions besides $\mathrm{a}=1$, b=3. If the problem is solved by enumeration - 1-7 points depending on the number of numbers enumerated. |