## SUBJECT II

Determine all pairs of natural numbers $(x, y)$ that satisfy the condition: $x^{3} \cdot y=1512-x^{3}$.
## SUBJECT II

Determine all pairs of natural numbers $(x, y)$ that satisfy the condition: $x^{3} \cdot y=1512-x^{3}$.

Grading criteria

$$
x^{3} \cdot y=1512-x^{3} \Leftrightarrow x^{3} \cdot y+x^{3}=1512 \Leftrightarrow x^{3}(y+1)=1512 .
$$

But $1512=1^{3} \cdot 1512=2^{3} \cdot 189=3^{3} \cdot 56=6^{3} \cdot 7$.

It follows that in the set of natural numbers

$$
\left(x^{3} ; y+1\right) \in\left\{\left(1^{3} ; 1512\right) ;\left(2^{3} ; 189\right) ;\left(3^{3} ; 56\right) ;\left(6^{3} ; 7\right)\right\}
$$

From which we have $(x ; y) \in\{(1 ; 1511) ;(2 ; 188) ;(3 ; 55) ;(6 ; 6)\} \quad 1.5 \text{ points}$