Arithmetic

First person will give food to every one else on the first day, then he will return back.

Second person will give food to everyone else remaining and then return back(as he is still left with food for 2 days)

Third person will give food to the rest and then will return back (he still have food for three days)

Similarly forth person will return back after 4th day

Now, the explorer has food and supplies for 6 days and he has to travel for 6 more days.

So, 5 will be sufficient.