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 $11$.

Let's determine which of the given numbers is a prime number. A prime number is a natural number greater than 1 that is only divisible by 1 and itself.

We have the following numbers to consider: $22, 23, 20,$ and $27$.

• 22: This number is divisible by 1, 2, 11, and 22. Therefore, it is not a prime number since it has divisors other than 1 and itself.
• 23: This number is only divisible by 1 and 23. It satisfies the definition of a prime number, as it has no divisors other than 1 and 23.
• 20: This number is divisible by 1, 2, 4, 5, 10, and 20. It has divisors other than 1 and itself, making it not a prime number.
• 27: This number is divisible by 1, 3, 9, and 27. Like the others, it is not a prime number because it has divisors other than 1 and itself.

Thus, after evaluating all options, we find that only $23$ is a prime number.

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

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 $11$.

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 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 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 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 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 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 $13$.

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 solve this problem, we will verify if each of the given numbers is a prime number. The numbers provided are $9$, $11$, $4$, and $6$.

• Option $9$: This number is not a prime because it has divisors other than 1 and itself; specifically, it is divisible by $3$ ($9 = 3 \times 3$).
• Option $11$: We need to check if $11$ is divisible by any number other than $1$ and $11$ itself, up to the square root of $11$ (approximately $3.32$). The integers to check are $2$ and $3$.

  $11$ is not even, thus not divisible by $2$.

  $11$ does not divide evenly by $3$ because $11 \div 3$ is not an integer.

  Since $11$ is not divisible by any number except $1$ and itself, it is a prime number.

• Option $4$: This number is not a prime because it is even, meaning it is divisible by $2$ ($4 = 2 \times 2$).
• Option $6$: This number is not a prime because it is divisible by $2$ and $3$ ($6 = 2 \times 3$).

Therefore, among the given options, the only prime number is $11$.

To solve this problem, we'll check each number to determine if it's a prime:

• Number 6: Greater than 1 but divisible by 2 and 3. Not prime.
• Number 4: Greater than 1 but divisible by 2. Not prime.
• Number 8: Greater than 1 but divisible by 2 and 4. Not prime.
• Number 2: Greater than 1 and not divisible by any other numbers except 1 and itself. It is prime.

Therefore, the prime number from the options is $2$.

To solve this problem, we will follow these detailed steps:

• Step 1: Understand the definitions: A composite number has more than two divisors. A prime number has exactly two divisors.
• Step 2: Examine each choice to determine its type:
• $9$: Check divisibility by numbers other than 1 and itself. $9$ is divisible by 3, as $9 \div 3 = 3$. Thus, 9 is composite.
• $7$: Check divisibility. Only divides evenly by 1 and 7. Thus, 7 is prime.
• $3$: Check divisibility. Only divides evenly by 1 and 3. Thus, 3 is prime.
• $5$: Check divisibility. Only divides evenly by 1 and 5. Thus, 5 is prime.
• Step 3: Conclude which number is composite based on the divisor check:
• Since 9 is the only number with more than two divisors (1, 3, and 9), it is the composite number.

Therefore, the solution to the problem is $9$.