Question 39
Let $a$, $b$, $c$, $d$ be pairwise coprime positive odd numbers. For positive integers $n$, we define
$$
f(n) = \lfloor \frac{n}{a} \rfloor + \lfloor \frac{n}{b} \rfloor + \lfloor \frac{n}{c} \rfloor + \lfloor \frac{n}{d} \rfloor.
$$
Prove that
$$
\sum_{n=1}^{abcd} (-1)^{f(n)} = 1.
$$
$$
f(n) = \lfloor \frac{n}{a} \rfloor + \lfloor \frac{n}{b} \rfloor + \lfloor \frac{n}{c} \rfloor + \lfloor \frac{n}{d} \rfloor.
$$
Prove that
$$
\sum_{n=1}^{abcd} (-1)^{f(n)} = 1.
$$