3. Prime factorization $15000=2^{3} \cdot 3 \cdot 5^{4}$
Solution of the equation: $6 x^{2}-5 x y+y^{2}=(2 x-y)(3 x-y)=0 \rightarrow\left[\begin{array}{l}y=2 x \\ y=3 x\end{array}\right.$

Case 1. $y=2 x$

$2 x+3 y=2 x+6 x=8 x$. The greatest divisor of $x$, for which $y=2 x$ is also a divisor, is $x_{1}=2^{2} \cdot 3 \cdot 5^{4}$, and the value of the expression $p_{1}=8 x_{1}=2^{5} \cdot 3 \cdot 5^{4}$

Case 2. $y=3 x$

$2 x+3 y=2 x+9 x=11 x$. The greatest divisor of $x$, for which $y=3 x$ is also a divisor, is $X_{2}=2^{3} \cdot 5^{4}$, and the value of the expression $p_{2}=11 x_{2}=11 \cdot 2^{3} \cdot 5^{4}$.

Comparison of $p_{1}$ and $p_{2}:$

$$
p_{1}-p_{2}=2^{5} \cdot 3 \cdot 5^{4}-11 \cdot 2^{3} \cdot 5^{4}=5^{4} \cdot 2^{3} \cdot(12-11)>0
$$

Answer: $p_{\max }=2^{5} \cdot 3 \cdot 5^{4}=60000$