Question 115
Let $n$ be a positive integer with exactly 36 distinct prime factors. For $k = 1, 2, 3, 4, 5$, let $c_k$ denote the number of integers in the interval $\left[\frac{(k-1)n}{5}, \frac{kn}{5}\right]$ that are coprime to $n$. Prove that
$$
\sum_{1 \le i \lt j \le 5} (c_i - c_j)^2 \ge 2^{36}.
$$
$$
\sum_{1 \le i \lt j \le 5} (c_i - c_j)^2 \ge 2^{36}.
$$