## Subject 3.

Determine how many natural numbers of the form $\overline{x y z}$ have the property that the number $\overline{x y z}-\overline{x y}-z$ is a perfect cube.

Collection Gazeta Matematică, series B

| Solution Details | Associated Scoring |
| :---: | :---: |
| We have $\overline{x y z}-\overline{x y}-z=10 \cdot \overline{x y}+z-\overline{x y}-z=9 \cdot \overline{x y}$. | 1p |
| Since $9=3^{2}$, it follows that $\overline{x y}=3^{3 k+1} \cdot m^{3}=3 \cdot\left(3^{k} \cdot m\right)^{3}$, where $k \in \mathbb{N}, m \in \mathbb{N}^{*}$. | $2 p$ |
| Since $10 \leq \overline{x y}=3 \cdot\left(3^{k} \cdot m\right)^{3} \leq 99$, it follows that $4 \leq\left(3^{k} \cdot m\right)^{3} \leq 33$. We obtain $k=0, m=2$, with $\overline{x y}=24$ or $k=0, m=3$ with $\overline{x y}=81$. For $k=0, m=1$ we have $\overline{x y}=81$. | $3 \mathbf{p}$ |
| We have $\overline{x y z} \in\{\overline{24 z} \mid z=\overline{0,9}\} \cup\{\overline{81 z} \mid z=\overline{0,9}\}$, in total 20 numbers. | $1 \mathbf{p}$ |
3. Determine how many natural numbers of the form $\overline{x y z}$ have the property that the number $\overline{x y z}-\overline{x y}-z$ is a perfect cube.