Question

Consider the numbers 13, 16, 27, and 41. a. Which of these numbers are prime? How do you know? b. Which of these numbers are composite? How do you know?(2 points/

Answer 1

To determine which of the numbers 13, 16, 27, and 41 are prime and which are composite, we first need to 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** is a natural number greater than 1 that has at least one positive divisor other than 1 and itself (i.e., it can be formed by multiplying two smaller natural numbers).

Now, let's evaluate each number:

**a. Prime Numbers:**
- **13:** The only divisors are 1 and 13. Thus, 13 is a prime number.
- **16:** The divisors are 1, 2, 4, 8, and 16. Since it has divisors other than 1 and itself, 16 is not a prime number.
- **27:** The divisors are 1, 3, 9, and 27. Since it has divisors other than 1 and itself, 27 is not a prime number.
- **41:** The only divisors are 1 and 41. Thus, 41 is a prime number.

**Conclusion for Prime Numbers: 13 and 41 are prime.**

**b. Composite Numbers:**
- **13:** As established, it is prime, so it is not composite.
- **16:** As it has divisors 1, 2, 4, 8, and 16, it is composite because it can be expressed as \(4 \times 4\) or \(2 \times 8\).
- **27:** As it has divisors 1, 3, 9, and 27, it is composite because it can be expressed as \(3 \times 9\).
- **41:** As established, it is prime, so it is not composite.

**Conclusion for Composite Numbers: 16 and 27 are composite.**

In summary:
- **Prime Numbers:** 13, 41
- **Composite Numbers:** 16, 27