patient intersection

We have a large set of medical claims, each of which is assigned an integer that uniquely identifies the claim as 
belonging to some otherwise anonymous person. (If multiple claims exist for the same person, they will have the same identifier) 
Each claim includes a diagnosis code that indicates what malady they suffer from. Suppose that we pivot these claims to produce a list of files, each named for the diagnosis, which would
contain a list of the integer patient identifiers who's medical claims indicated that they were diagnosed with that
condition.

Now, given that list of files, we want to find the number of people who suffer from BOTH migraines (G43) and gastric ulcers (K25):

G43.txt:
3453234
6367323
8826474
4734737
...


K25.txt:
3453434
4577443
2367733
4574746
...

How should we go about finding the answer to our question? What if these files contain too many patients to fit in memory?
How could we optimize this process for speed? How could we optimize for memory usage? 
Please indicate the runtime complexity of your solution.