How many `5`-digit prime numbers can be formed using the digits `1, 2, 3, 4, 5` if the repetition of digits is not allowed?

Options

A. `5`
B. `4`
C. `3`
D. `0`

Answer

To find out how many 5-digit prime numbers can be formed using the digits 1, 2, 3, 4, and 5 without repetition, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the digits available**: We have the digits 1, 2, 3, 4, and 5.

2. **Calculate the sum of the digits**:
\[ 1 + 2 + 3 + 4 + 5 = 15 \]
The sum of these digits is 15.

3. **Check divisibility by 3**:
According to the divisibility rule of 3, if the sum of the digits of a number is divisible by 3, then that number is also divisible by 3. Since 15 is a multiple of 3, any number formed using all these digits will also be divisible by 3.

4. **Determine the nature of prime numbers**:
A prime number is defined as a number greater than 1 that has no positive divisors other than 1 and itself. Therefore, a prime number cannot be divisible by 3 (except for the prime number 3 itself).

5. **Conclusion**: Since any 5-digit number formed by the digits 1, 2, 3, 4, and 5 will be divisible by 3, none of these numbers can be prime.

Thus, the total number of 5-digit prime numbers that can be formed using the digits 1, 2, 3, 4, and 5 without repetition is **0**.

### Final Answer:
0