Let n, k be positive integers. Prove that \(\sum_{i=1}^n k^{\gcd(i,n)}\) is divisible by n.

Let 1≤a₁<a₂<…<a_k<n be a sequence of integers such that gcd(a_i,a_j)=1 for all 1≤i<j≤k. What is the maximum value of k?

(Problem updated on Sep. 26, 8AM: gcd(a_i,a_j)=1.)