Prime Numbers 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 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 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 $n = 20$ 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 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.