Checking whether a number divides any of a set of other numbers
A silly answer:

Precompute the union of the sets of the factors of all elements in the set.

Then checking if the number is a divisor of any elements in the set is just a single set lookup ($O(1)$). This, however, requires an absurd amount of precomputation (and space to store the resulting set).

The precomputation time can probably be improved - sort the input set, then for each number from 1 to the square root of the largest number in the set, check if any of the numbers from the number squared to the largest number in the set are divisible by the number (add it to the cache if so). It's still absurd, however.