CMO 2021 Problem 3

Let $n$ be positive integer such that there are exactly 36 different prime numbers that divides $n.$ For $k=1,2,3,4,5,$ $c_n$ be the number of integers that are mutually prime numbers to $n$ in the interval $[\frac{(k-1)n}{5},\frac{kn}{5}] .$ $c_1,c_2,c_3,c_4,c_5$ is not exactly the same.Prove that $$\sum_{1\le i<j\le 5}(c_i-c_j)^2\geq 2^{36}.$$