Mathematical Properties: Prime Number Detection Exercise

To determine which number is a prime number among the choices, we proceed with the following analysis:

First, let's define a prime number:
A prime number is a number greater than 1 that has no divisors other than 1 and itself.

• Check $34$: $34$ is even, hence divisible by 2. So, it is not a prime number.
• Check $30$: $30$ is also even and divisible by 2. Thus, it is not a prime number.
• Check $32$: $32$ is even, so it is divisible by 2. Therefore, it is not a prime number.
• Check $31$: Let's check for divisibility by numbers up to the square root of $31$, which is approximately 5.57.
• It is not divisible by 2 because it is not even.
• It is not divisible by 3, as $31 \div 3$ does not result in an integer.
• It is not divisible by 5, as it does not end in 0 or 5.
• Since $31$ is not divisible by any prime number up to its square root, $31$ is a prime number.

Therefore, the solution to the problem is $31$, which is a prime number.