Let S be a set of 6 integers taken from {1, 2, ..., 12} with the prope

Let’s start with 2 as the smallest number, as 1 is not among the answer choices. We can have 2, 3, 5, 7, 11 in our set. However, since any other number we will add will either be a multiple of 2 or a multiple of 3, this set of numbers does not work. Also, there are no numbers in the set which will enable us to add more than one number if we don’t include (for instance, if we don’t add 3, we will be able to add 9 but we still have five numbers. If we don’t include 5, we will be able to add 10 but we still have five numbers).

Now, let’s try 3 as the smallest number. We can have 3, 4, 5, 7, 11. However, again any other number we add will be a multiple of a number that is already in the set (6 is a multiple of 3, 8 is a multiple of 4, 10 is a multiple of 5 etc.) Again there are no numbers which would make it possible to add more than one number if not included, similar to the above case.

Let’s try 4 as the smallest number. We can have 4, 5, 6, 7, 9, 11. We see that we have followed the rules for the set, so 4 is the smallest possible number.

Answer: C