858. Find the smallest pair of natural numbers $x$ and $y$ that satisfy the equation

$$
5 x^{2}=3 y^{5}
$$
$\triangle$ From the equation, it is clear that $x$ is divisible by 3, and $y$ is divisible by 5:

$$
x=3 x_{1}, \quad y=5 y_{1},
$$

where $x_{1}$ and $y_{1}$ are natural numbers. Then

$$
5 \cdot 3^{2} x_{1}^{2}=3 \cdot 5^{5} y_{1}^{5}, \quad 3 x_{1}^{2}=5^{4} y_{1}^{5}
$$

From this, it follows that $x_{1}$ is divisible by 5; moreover, $y_{1}$ is divisible by 3, and in this case, $x_{1}$ is also divisible by 3:

$$
x_{1}=15 x_{2}, \quad y_{1}=3 y_{2}
$$

where $x_{2}$ and $y_{2}$ are natural numbers. We will have:

$$
3 \cdot 15^{2} x_{2}^{2}=5^{4} \cdot 3^{5} y_{2}^{5}, \quad x_{2}^{2}=3^{2} \cdot 5^{2} y_{2}^{5}
$$

Therefore, $y_{2}^{5}$ must be a perfect square. The smallest $y_{2}$ that satisfies this condition is $y_{2}=1$. Then

$$
x_{2}^{2}=15^{2}, \quad x_{2}=15
$$

It remains to find $x_{1}, y_{1}, x$, and $y$:

$$
x_{1}=15 x_{2}=15^{2}=225, \quad y_{1}=3 y_{2}=3 ; \quad x=3 x_{1}=3 \cdot 225=675, \quad y=5 y_{1}=15
$$

Answer: $x=675, y=15$.