Prime Numbers and Composite Numbers

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 if the number $n = 4$ is prime or composite, we will follow these steps:

• Step 1: Understand the definitions.
  A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A composite number has additional divisors.
• Step 2: Identify divisors of 4.
  We list out the divisors of 4, starting from 1: They are 1, 2, and 4.
• Step 3: Analyze the divisors.
  The number 4 has more than two divisors: 1, 2, and 4. This means it can be divided by numbers other than 1 and itself.

Conclusion: Since 4 has divisors other than 1 and itself (specifically, it is divisible by 2), it is not a prime number. Therefore, 4 is classified as a composite number.

Therefore, the solution to the problem is Composite.

To determine whether the number $n = 7$ is prime or composite, we follow these steps:

• Step 1: Acknowledge the definition of prime numbers. A prime number is any number greater than 1 that has no divisors other than 1 and itself.
• Step 2: We begin by checking if the number $7$ is greater than 1. Since $7 > 1$, it is eligible to be considered a prime number.
• Step 3: We examine whether $7$ has any divisors other than 1 and itself.
• Step 4: For a number to be composite, it must have additional divisors apart from 1 and itself. Let's check the possible divisors.
• Step 5: Since 7 is a small number, its divisors would be smaller than $\sqrt{7} \approx 2.64$. The only whole number less than or equal to 2 not including 1 is 2.
• Step 6: We check divisibility: 7 divided by 2 is not a whole number, confirming 7 is not divisible by any number other than 1 and itself.

Therefore, we conclude that the number $n = 7$ is indeed a Prime number.

To determine whether the number $n = 8$ is prime or composite, we will use the definitions of prime and composite numbers:

• A prime number is a natural number greater than 1 that has no divisors other than 1 and itself.
• A composite number is a natural number greater than 1 that has divisors other than 1 and itself.

Let's analyze $n = 8$:

Step 1: Since 8 is greater than 1, it can be either prime or composite.

Step 2: List the divisors of 8. The divisors of 8 are 1, 2, 4, and 8.

Step 3: Verify if 8 has divisors other than 1 and itself. We see that 8 is divisible by 2 and 4, in addition to 1 and 8.

Since 8 has divisors other than 1 and itself, 8 is not a prime number.

Therefore, 8 is classified as a composite number.

Thus, the correct answer is composite.

To solve this problem, we'll determine whether $n = 22$ is a prime or composite number.

We follow these steps:

• Step 1: List possible divisors of $22$ other than $1$ and $22$ itself.
• Step 2: Test $22$ for divisibility by these numbers.
• Step 3: Conclude based on the results.

Step 1: The numbers to consider are $2, 3, 4, 5, ...$ up to the square root of $22$, rounded up, which is approximately 4.7. Thus, feasible numbers are $2, 3, 4$.

Step 2: Check each number:

• $\text{Is } 22 \div 2$ a whole number? Yes, $22 \div 2 = 11$.

Step 3: Since $22$ is divisible by $2$, it has at least one divisor other than $1$ and itself.

Therefore, $n = 22$ is a composite number.

Thus, the correct choice from the given options is: Composite.