Prime Numbers and Composite Numbers: Identifying the prime number

To solve this problem, we'll identify which of the given numbers is a prime number:

• Step 1: Define a prime number as a positive integer greater than 1 that has no divisors other than 1 and itself.
• Step 2: Examine each number and list its divisors.

Now, let's work through each step:

Step 1: Consider the numbers given: $9$, $11$, $8$, and $4$.

Step 2:

• $9$ has divisors $1, 3, 9$. Since it has more than two divisors, it is not a prime number.
• $11$ has divisors $1, 11$. Since it has exactly two divisors, it is a prime number.
• $8$ has divisors $1, 2, 4, 8$. Since it has more than two divisors, it is not a prime number.
• $4$ has divisors $1, 2, 4$. Since it has more than two divisors, it is not a prime number.

Therefore, the number that is a prime number is $11$.

To determine which of the given numbers is a prime number, we need to check each number for divisibility:

• For $9$: It is divisible by $3$ (since $9 \div 3 = 3$), so it is not a prime number.

• For $10$: It is divisible by $2$ (since $10 \div 2 = 5$), so it is not a prime number.

• For $7$: It is only divisible by $1$ and $7$ (itself). It cannot be divided by any other numbers except 1 and itself without leaving a remainder, so $7$ is a prime number.

• For $12$: It is divisible by $2$ (since $12 \div 2 = 6$), so it is not a prime number.

Thus, the only number in the list that satisfies the condition of being prime, having exactly two distinct positive divisors, is $7$.

Therefore, the solution to the problem is $7$.

To solve this problem, we'll follow these steps:

• Step 1: Identify the given numbers: $8$, $4$, $5$, $12$.
• Step 2: Verify the primality of each number.

Now, let's work through each step:
Step 1: The numbers provided are $8$, $4$, $5$, $12$.
Step 2: We need to determine if each number is a prime by checking if they have divisors other than 1 and themselves:

• $8$ is divisible by $2$, so it has divisors besides 1 and 8. Therefore, 8 is not a prime number.
• $4$ is divisible by $2$, so it has divisors besides 1 and 4. Therefore, 4 is not a prime number.
• $5$ is not divisible by any integer other than 1 and 5. Thus, 5 is a prime number.
• $12$ is divisible by $2$ and $3$, so it has divisors besides 1 and 12. Therefore, 12 is not a prime number.

Therefore, the solution to the problem is $5$.

To solve this problem, we'll verify whether each number is a prime number:

• 22: Divisible by 2. Not a prime number.

• 23: Divisible only by 1 and 23. It is a prime number.

• 21: Divisible by 3 and 7. Not a prime number.

• 24: Divisible by 2. Not a prime number.

Now, let's go through each step in detail:

Step 1: Check 22
22 is even, meaning it is divisible by 2. As it has divisors other than 1 and itself, 22 is not a prime number.

Step 2: Check 23
Begin by testing divisibility by 2. Since 23 is odd, it is not divisible by 2. Next, check divisibility by 3: $23 \div 3 \approx 7.67$, not an integer. No divisors other than 1 and 23 are found, therefore 23 is a prime number.

Step 3: Check 21
21 is odd, so not divisible by 2. However, $21 \div 3 = 7$, which is an integer. Thus, 21 is not prime because it is divisible by numbers other than 1 and itself.

Step 4: Check 24
24 is even and divisible by 2. Hence, 24 is not a prime number.

Therefore, the solution to the problem is $23$.

To determine which number is a prime number, we will check each choice for primality based on whether it has divisors other than 1 and itself:

• Check $16$: This number is even and divisible by 2, hence not prime.
• Check $17$: Testing divisibility by integers up to the square root of 17 (roughly 4.1), we find no divisors other than 1 and $17$ itself. Therefore, $17$ is prime.
• Check $18$: This number is even and divisible by 2, hence not prime.
• Check $14$: This number is also even and divisible by 2, hence not prime.

Given these observations, the only prime number among the choices is $17$.

To determine which of the given numbers is a prime number, follow these steps:

• Step 1: Check the number 39.
  39 is divisible by 3 (since 3 + 9 = 12, which is divisible by 3), so it has divisors other than 1 and itself, and thus 39 is not prime.
• Step 2: Check the number 34.
  34 is divisible by 2 (as it's an even number), so it has divisors other than 1 and itself, and thus 34 is not prime.
• Step 3: Check the number 42.
  42 is divisible by 2 (as it's an even number), so it has divisors other than 1 and itself, and thus 42 is not prime.
• Step 4: Check the number 37.
  For a number to be prime, it should not be divisible by any numbers other than 1 and itself. We check divisibility of 37 by numbers up to its square root (~6.08), which are 2, 3, and 5.
• 37 is not divisible by 2 (as it is odd).
• 37 is not divisible by 3 (as 3 + 7 = 10, which is not divisible by 3).
• 37 is not divisible by 5 (as it does not end in 0 or 5).

Since 37 is not divisible by any integer other than 1 and 37 itself, it is a prime number.

Therefore, the solution to the problem is $37$.

To determine which of the given numbers is a prime number, we will evaluate each one to check if it has any divisors other than 1 or itself.

Here are the steps:

• Step 1: Examine number 18:
  - 18 is divisible by 2 (as it is even) and also by 3 (since 1 + 8 = 9, which is divisible by 3). Therefore, 18 is not a prime number.
• Step 2: Examine number 21:
  - 21 is divisible by 3 (since 2 + 1 = 3, which is divisible by 3) and by 7 (as $21 \div 7 = 3$). Thus, 21 is not a prime number.
• Step 3: Examine number 16:
  - 16 is divisible by 2 (multiple times: 16, 8, 4, 2) and hence is not a prime number.
• Step 4: Examine number 19:
  - Check divisibility by 2: 19 is odd, not divisible by 2.
  - Check divisibility by 3: 1 + 9 = 10, not divisible by 3.
  - Check divisibility by 5: Does not end in 0 or 5.
  - Check divisibility by any prime numbers less than the square root of 19: None divides 19 evenly.
  As no number other than 1 and itself divides 19, it is a prime number.

Therefore, the solution is that $19$ is the prime number among the choices given.

To determine which number is a prime number, we must individually verify each of the options provided.

• Check $57$:
• Divisibility by 2: $57$ is odd, so not divisible.
• Divisibility by 3: Sum of digits $5 + 7 = 12$, divisible by 3, hence 57 is divisible by 3.
• Conclusion: $57$ is not a prime number.
• Check $58$:
• Divisibility by 2: $58$ is even, hence divisible by 2.
• Conclusion: $58$ is not a prime number.
• Check $55$:
• Divisibility by 2: $55$ is odd, not divisible by 2.
• Divisibility by 3: Sum of digits $5 + 5 = 10$, not divisible by 3.
• Divisibility by 5: Ends with 5, divisible by 5.
• Conclusion: $55$ is not a prime number.
• Check $60$:
• Divisibility by 2: $60$ is even, hence divisible by 2.
• Conclusion: $60$ is not a prime number.

After evaluating all options, none of the provided numbers is a prime number. It appears there may be a misunderstanding in the problem or typo, as no addtional information is given to explain an alternative solution.

Thus, none of the above numbers are prime, and we should conclude there is either a mistake in the given problem or choices.

To determine which of the provided numbers is a prime number, follow these steps:

• Step 1: Understand the definition of a prime number. A prime number is a number greater than 1 that has no positive divisors other than 1 and itself.
• Step 2: Analyze and test each choice for primality:
• 45: Check divisibility by 2 (not even), by 3 (4 + 5 = 9, divisible by 3). Therefore, 45 is not prime.
• 46: Check divisibility by 2 (46 is even). Therefore, 46 is not prime.
• 42: Check divisibility by 2 (42 is even). Therefore, 42 is not prime.
• 43: Check divisibility by 2 (not even), by 3 (4 + 3 = 7, 7 is not divisible by 3), by 5 (does not end in 0 or 5). No divisors are found for 43. Hence, it is prime.

Upon checking, we see that 43 is the only number that cannot be evenly divided by any number other than 1 and itself. Thus, it is a prime number.

Therefore, the solution is $43$.

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.

To solve this problem, we identify which of the given numbers is a prime number.

Let's evaluate each provided number:

• $13$: This number is only divisible by 1 and 13 itself. Therefore, 13 is a prime number.

• $12$: This number is divisible by 1, 2, 3, 4, 6, and 12. Since it has divisors other than 1 and itself, it is not prime.

• $15$: This number is divisible by 1, 3, 5, and 15. Since it has divisors other than 1 and itself, it is not prime.

• $4$: This number is divisible by 1, 2, and 4. Since it has divisors other than 1 and itself, it is not prime.

Therefore, the only prime number among the choices is $13$.

Test for divisibility:

• $6$: It is divisible by $2$ and $3$, so it is not a prime number.

• $11$: It is only divisible by $1$ and $11$, making it a prime number.

• $10$: It is divisible by $2$ and $5$, so it is not a prime number.

• $4$: It is divisible by $2$, so it is not a prime number.

Based on the steps above, the number that is a prime number is $11$.