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 correct option is Zero.

To determine how many five-digit prime numbers can be formed using the digits 1, 2, 3, 4, and 5 without repetition, we can analyze the divisibility properties of any number formed by these digits.

First, let us calculate the sum of the given digits:

$\text{Sum of digits}=1+2+3+4+5=15$

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 is 15, and 15 is divisible by 3 ($15÷3=5$), any five-digit number formed using all of these digits (regardless of their arrangement) will always have a digit sum of 15 and will therefore be divisible by 3.

Furthermore, because the number is a five-digit number, it is strictly greater than 3. Any integer greater than 3 that is divisible by 3 is a composite number, not a prime number.

Thus, it is impossible to form any prime numbers using all five digits 1, 2, 3, 4, and 5 without repetition. The total number of such five-digit prime numbers is Zero.