How many five-digit prime numbers can be obtained by using all the digits 1, 2, 3, 4 and 5 without repetition of digits?

The sum of the digits 1, 2, 3, 4, and 5 is 1 + 2 + 3 + 4 + 5 = 15. A fundamental divisibility rule states that a number is divisible by 3 if the sum of its digits is divisible by 3. Since the sum of the digits (15) is divisible by 3, any five-digit number formed by using these digits without repetition will also be divisible by 3. For a number to be prime, it must be a natural number greater than 1 that has no positive divisors other than 1 and itself. Any number formed using these digits (e.g., 12345, 54321, etc.) will be a five-digit number, which is clearly greater than 3. Since all such numbers are divisible by 3 and are greater than 3, they have 3 as a divisor in addition to 1 and themselves. Therefore, none of these numbers can be prime. Thus, the number of five-digit prime numbers that can be obtained using these digits is 0. Analyzing the options: A) 0: This is correct because no such prime number can be formed. B) 1: Incorrect, as explained above. C) 9: Incorrect. D) 10: Incorrect. The final answer is A.