Core Idea
The answer is a time. If the machines can produce enough items by time t, then any larger time also works.
Algorithm
Binary search the smallest feasible time. For a fixed time, sum floor(t / machineTime) over all machines, stopping early once the target is reached.
Common Mistakes
Use 64-bit integers and choose a high bound that is definitely feasible, such as the fastest machine time multiplied by the target count.