This problem can be efficiently solved using binary search on the possible time range. The minimum possible time is 0, and a safe upper bound is the maximum response time multiplied by the total number of requests (max(servers) * k).
For a given mid time during the binary search, we can calculate the total number of requests that can be processed by all servers within that mid time. For each server, the number of requests it can handle is mid // server_response_time. Summing this value across all servers gives us the total requests processed.
If the total requests processed is greater than or equal to k, it means mid is a possible completion time, and we try a smaller time by setting right = mid - 1. Otherwise, if fewer than k requests can be processed, we need more time, so we set left = mid + 1.
The binary search continues until left exceeds right, and the final left value will be the minimum time required to process k requests.
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