Let X be a four - digit positive integer such that the unit digit of X is prime and 

the product of all digits of X is also prime. How many such integers are possible?

Given that unit digit of $$X$$ and product of all 4 digits of $$X$$ are prime.

The product of two numbers to be prime is possible only when one of the numbers is prime and the other is ‘1’.

The possibilities for the prime unit digits are - 2,3,5,7     (Note that 1 is not a prime number)

Hence the possibility of remaining 3 digits, considering the product of all 4 digits to be prime is ‘111’ only.

Hence all the possible numbers are 1112,1113,1115,1117

$$\therefore$$ Total 4 integers are possible.