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?

(A) 4

(B) 8

(C) 12

(D) 24

(E) None of the above
Solution follows here:

Solution:

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 ---- total 4 possibilities

Answer(A)
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