Question 36
A sequence of positive integers $a_{1}, a_{2}, a_{3}, \ldots$ satisfies
$$
a_{n+1}=n\left\lfloor\frac{a_{n}}{n}\right\rfloor+1
$$
for all positive integers $n$. If $a_{30}=30$, how many possible values can $a_{1}$ take? (For a real number $x$, $\lfloor x\rfloor$ denotes the largest integer that is not greater than $x$.)
$$
a_{n+1}=n\left\lfloor\frac{a_{n}}{n}\right\rfloor+1
$$
for all positive integers $n$. If $a_{30}=30$, how many possible values can $a_{1}$ take? (For a real number $x$, $\lfloor x\rfloor$ denotes the largest integer that is not greater than $x$.)