For some positive integers $m > n$, the quantities $a = \text{lcm}(m, n)$ and $b = \gcd(m, n)$ satisfy $a = 30b$. If $m - n$ divides $a$, then what is the value of $\frac{m + n}{b}$?

Let's think step by step and output the final answer within \boxed{}.
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