Question

Let X be a four-digit positive integer such that the unit digit of X is prime and the product of all 4 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

Hint:

If a product of digits is prime, then all digits except one must be 1, because prime numbers have only one non-trivial factor.

Answer 1

The Correct Option is A

Step 1: Condition on unit digit.
Unit digit must be prime. So possible unit digits = 2, 3, 5, 7.
Step 2: Condition on product of all 4 digits.
Product of digits is prime. A prime number can only be obtained if one of the digits is prime and the other three digits are all 1. This is because product = prime × 1 × 1 × 1 = prime.
Step 3: Form numbers.
The first three digits must all be 1. The last digit (unit digit) must be prime. Hence possible numbers = 1112, 1113, 1115, 1117.
Step 4: Count.
Thus, there are exactly 4 such numbers. \[ \boxed{4} \]