Solution:

The equations say $x \cdot \gcd(x,y)^2 = 10^{60}$ and $y \cdot \operatorname{lcm}(x,y)^2 = 10^{570},$ so $x$ and $y$ are products of the primes $2$ and $5$ only. Fix one of these primes and let $a$ and $b$ be its exponents in $x$ and $y.$ Since the gcd takes the smaller exponent and the lcm the larger, $$a + 2\min(a, b) = 60, \qquad b + 2\max(a, b) = 570.$$

If $a \gt b,$ then $a + 2b = 60$ and $b + 2a = 570;$ adding gives $a + b = 210,$ and subtracting gives $a - b = 510,$ forcing $b \lt 0,$ impossible. So $a \le b,$ and the equations become $3a = 60$ and $3b = 570,$ giving $a = 20$ and $b = 190$ for both primes.

Thus $x = 2^{20} 5^{20}$ and $y = 2^{190} 5^{190},$ so $m = 40,$ $n = 380,$ and $3m + 2n = 120 + 760 = 880.$