Question 114
For any integer $n$, define $\lfloor n\rfloor$ as the greatest integer less than or equal to $n$. For any positive integer $n$, let
$$
f(n) = \lfloor n \rfloor + \left\lfloor \frac{n}{2} \right\rfloor + \left\lfloor \frac{n}{3} \right\rfloor + \cdots + \left\lfloor \frac{n}{n} \right\rfloor.
$$
For how many values of $n$, $1 \leq n \leq 100$, is $f(n)$ odd?
$$
f(n) = \lfloor n \rfloor + \left\lfloor \frac{n}{2} \right\rfloor + \left\lfloor \frac{n}{3} \right\rfloor + \cdots + \left\lfloor \frac{n}{n} \right\rfloor.
$$
For how many values of $n$, $1 \leq n \leq 100$, is $f(n)$ odd?