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