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 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.