Prime Numbers and Composite Numbers: Recognizing prime and composite numbers

A number is classified as prime if it has exactly two distinct positive divisors: 1 and itself. Conversely, a number is composite if it has more than two divisors.

Given the number $n = 10$, we need to determine whether it is prime or composite.

Let's test the divisibility of 10 by numbers other than 1 and 10:

• Check divisibility by 2: Since 10 is an even number, it is divisible by 2. Specifically, $10 \div 2 = 5$ with no remainder.
• Check divisibility by 3: $10 \div 3 \approx 3.333$, which is not an integer, hence not divisible.
• Check divisibility by 5: $10 \div 5 = 2$ with no remainder, showing divisibility by 5.

Since 10 is divisible by numbers other than 1 and itself (specifically 2 and 5), it is not prime. Therefore, the number 10 is composite.

In conclusion, the number 10 is a composite number.

To determine if $n = 20$ is prime or composite, we need to examine its divisors.

• Step 1: Identify divisors of 20, other than 1 and 20 itself.
  Since 20 is an even number, it is divisible by 2. So, 2 is a divisor of 20.
• Step 2: Check divisibility by other small numbers:
  - 20 divided by 4 equals 5 (another divisor).
  - Additionally, $20 \div 5 = 4$, confirming 5 is also a divisor.

Since 20 has divisors other than 1 and itself (including 2, 4, and 5), it is not a prime number.

Therefore, the number $n = 20$ is Composite.

To determine if the number 19 is prime, follow these steps:

• Step 1: Check if the number is greater than 1. Since $19 > 1$, proceed to the next step.
• Step 2: Identify potential divisors for 19 by considering integers from 2 up to $\lfloor \sqrt{19} \rfloor$.

The square root of 19 is approximately 4.36, and thus we test divisibility by integers 2, 3, and 4.

• 19 divided by 2: The quotient is not an integer (it gives 9.5).
• 19 divided by 3: The quotient is not an integer (it gives 6.333...).
• 19 divided by 4: The quotient is not an integer (it gives 4.75).

None of these divisions result in an integer, meaning 19 has no divisors other than 1 and 19 itself.

Therefore, the number 19 is prime.

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 36 is a prime or composite number, we need to check if it has divisors other than 1 and 36:

• Step 1: Calculate the square root of 36, which is 6. This means we only need to test divisibility by numbers up to and including 6.
• Step 2: Check divisibility by 2. Since 36 is an even number (divisible by 2), it has a divisor other than 1 and itself.

Therefore, since 36 is divisible by 2 (and also by other numbers such as 3, 4, and more), it has divisors other than just 1 and 36. This means it cannot be a prime number.

Conclusively, the number 36 is Composite.

To determine whether $n = 17$ is a prime number, we will check if it has any divisors other than 1 and itself. A prime number has no divisors other than these two.

• Step 1: Calculate $\sqrt{17}$. Since $\sqrt{17} \approx 4.123$, we need to test divisibility by integers 2, 3, and 4.
• Step 2: Check divisibility by 2. Since 17 is odd, it is not divisible by 2.
• Step 3: Check divisibility by 3. The sum of the digits of 17 is $1 + 7 = 8$, which is not divisible by 3.
• Step 4: Check divisibility by 4. Half of 17 is not a whole number, thus it is not divisible by 4.

Since 17 is not divisible by any number other than 1 and itself, it satisfies the condition of being a prime number.

Therefore, the solution to the problem is Prime.

To determine whether $n = 23$ is a prime number, we will test its divisibility:

• Step 1: Calculate $\sqrt{23}$. The approximate value is 4.795, and thus we consider prime numbers up to the integer part, which is 4.
• Step 2: Check if 23 is divisible by any prime numbers less than or equal to 4. These primes are 2 and 3.

Step 3: Test divisibility:
- 23 is not divisible by 2, as it is odd.
- 23 is not divisible by 3, since $23 \div 3 \approx 7.67$, which is not an integer.

Since 23 is not divisible by any prime number less than or equal to its square root, it only has divisors of 1 and 23. Hence, 23 is a prime number.

Therefore, the solution to the problem is that $n = 23$ is prime.

To determine whether the number $n = 29$ is prime or composite, we will check if $n$ has any divisors other than 1 and itself.

The definition of a prime number is one that has exactly two distinct positive divisors: 1 and itself. Conversely, a composite number has more than two distinct positive divisors.

First, observe that the square root of 29 is approximately 5.385. This tells us that we only need to check divisibility by all prime numbers less than or equal to 5. These primes are 2, 3, and 5.

• Check divisibility by 2: 29 is odd, so it is not divisible by 2.
• Check divisibility by 3: The sum of the digits of 29 is $2 + 9 = 11$, which is not divisible by 3, so 29 is not divisible by 3.
• Check divisibility by 5: 29 does not end in 0 or 5, so it is not divisible by 5.

Since 29 is not divisible by any of these primes, it has no divisors other than 1 and 29 itself. Therefore, 29 is a prime number.

Hence, the solution to the problem is that the number $n = 29$ is Prime.

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 if 42 is a prime or composite number by checking its divisibility by numbers other than 1 and itself.

A number is prime if it has exactly two distinct positive divisors: 1 and itself. It is composite if it has more than two distinct divisors.

Let's find the divisors of 42:

• $42 \div 1 = 42$
• $42 \div 2 = 21$ (evenly divisible, so 2 is a divisor)
• $42 \div 3 = 14$ (evenly divisible, so 3 is also a divisor)
• $42 \div 6 = 7$ (evenly divisible, so 6 is another divisor)
• $42 \div 7 = 6$ (evenly divisible, so 7 is a divisor)
• $42 \div 14 = 3$ (evenly divisible, so 14 is a divisor)
• $42 \div 21 = 2$ (evenly divisible, so 21 is a divisor)
• $42 \div 42 = 1$

From the above list, we can see that 42 has divisors other than 1 and itself, namely 2, 3, 6, 7, 14, and 21. This means that 42 is not a prime number.

Therefore, the number 42 is a composite number.

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.

To solve this problem, we'll determine if the number $n = 14$ is prime or composite by checking its divisibility by numbers other than 1 and 14 itself.

• Step 1: Recognize that a prime number has only two distinct divisors: 1 and itself.
• Step 2: A composite number has additional divisors.
• Step 3: Start checking for divisibility from the smallest primes: 2, 3, 5, etc.

Now, let's work through these steps:

Step 1: Given $n = 14$, check divisibility:
Since 14 is an even number, it is divisible by 2 (i.e., $14 \div 2 = 7$).

Step 2: As 2 is a divisor other than 1 and itself, 14 cannot be a prime number.

Step 3: Verify divisibility confirms that 14 is composed of factors $2 \times 7$.

Therefore, the solution to the problem is that the number 14 is composite.

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

• Step 1: Verify if the number $n = 11$ is divisible by any numbers other than 1 and itself.
• Step 2: Based on the result of the divisibility test, determine the type of number.

Now, let's work through each step:

Step 1: Check divisibility.
The number 11 is greater than 2, so we check divisibility by smaller primes up to $\sqrt{11} \approx 3.3$. The prime numbers less than or equal to 3 are 2 and 3.

- **Divisibility by 2:** 11 is an odd number, hence not divisible by 2.
- **Divisibility by 3:** Sum of digits of 11 is $1 + 1 = 2$, which is not divisible by 3.

Step 2: Conclusion based on divisibility.
Since 11 is not divisible by any other numbers except 1 and 11 itself, it does not have any divisors other than 1 and itself. Therefore, 11 is a prime number.

Hence, the solution to the problem is $n = 11$ is Prime.