Let $$$f(x) = (x \bmod p) \bmod q$$$ and $$$g(x) = (x \bmod q) \bmod p$$$.

Let $$$A(l, r) = \min(\sum \limits_{l \le i \le r} f(a_i), \sum \limits_{l \le i \le r} g(a_i))$$$ and $$$B(l, r) = \sum \limits_{l \le i \le r} min(f(a_i), g(a_i))$$$.

The answer is $$$\min \limits_{i=1}^{n-k+1} (A(i, i+k-1) + B(1, i-1) + B(i+k, n))$$$.