Some remarkable math

It is best to start with some good math.

It is true that 2^p-2 is always a whole multiple of p for any prime number p. For example if p=5 then 2⁵-2 = 32-2 = 30 = 6 times 5 which is a multiple of 5. Similarly if p=7 then 2⁷-2 = 128-2 = 126 = 18 times 7 which is a multiple of 7. However if we take the nonprime p=6 we get 2⁶-2 = 64-2 = 62 which is not a multiple of 6.

Question 1: Does this work if we replace 2 by other integers such as 3? (In which case we would we would be asking whether 3^p-3 is a multiple of p.)

Questions 2: We know that the result does not hold for the nonprime p=6. But could there be any nonprime p for which this works?