Correct Option (A)
The given digits are 1, 2, 3, 4, and 5. To form a five-digit number using these digits without repetition, we first determine the sum of these digits: 1 + 2 + 3 + 4 + 5 = 15.
According to the divisibility rule for 3, 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 number formed by arranging these digits will also be divisible by 3.
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. Since any five-digit number formed using these digits will be greater than 3 and divisible by 3, it will have at least three divisors (1, 3, and the number itself). Therefore, none of these numbers can be prime.
Thus, the number of five-digit prime numbers that can be obtained is zero.
Incorrect Options:
Options (B) One, (C) Nine, and (D) Ten are incorrect. As established, any five-digit number formed using the digits 1, 2, 3, 4, and 5 without repetition will be divisible by 3 and greater than 3, which means none of them can be a prime number.