Question 87
Call a positive integer $n$ quixotic if the value of
$$
\operatorname{lcm}(1,2,3, \ldots, n) \cdot\left(\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{n}\right)
$$
is divisible by $45$. Compute the tenth smallest quixotic integer.
$$
\operatorname{lcm}(1,2,3, \ldots, n) \cdot\left(\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{n}\right)
$$
is divisible by $45$. Compute the tenth smallest quixotic integer.