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Chapter 5: Problem 30

Determine whether each number is prime, composite, or neither. \([2.1 \mathrm{c}]\) $$ 28 $$

Short Answer

Expert verified
28 is a composite number.

Step by step solution

01

Understand Prime and Composite Numbers

Prime numbers are numbers greater than 1 with only two positive divisors: 1 and itself. Composite numbers have more than two positive divisors. Numbers less than 2 are neither prime nor composite.
02

Check Divisibility

To determine if 28 is prime or composite, check if it has any divisors other than 1 and 28.
03

Find Divisors of 28

List the positive divisors of 28: 1, 2, 4, 7, 14, and 28.
04

Determine Type of Number

Since 28 has more than two positive divisors, it is a composite number.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Prime Numbers
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. In simpler terms, prime numbers cannot be divided evenly by any other number except for 1 and the number itself. For instance, consider the number 5: its only divisors are 1 and 5, making it a prime number.

To identify a prime number:
  • Check if the number is greater than 1.
  • Verify if it has exactly two distinct positive divisors.
Some examples of prime numbers include 2, 3, 5, 7, and 11. Note that 2 is the only even prime number. All other even numbers can be divided by 2, making them composite.
Composite Numbers
A composite number is a natural number greater than 1 that has more than two positive divisors. In other words, composite numbers can be divided evenly by numbers other than 1 and themselves. For instance, the number 28 has the divisors 1, 2, 4, 7, 14, and 28.

To identify a composite number:
  • Check if the number is greater than 1.
  • Verify if it has more than two distinct positive divisors.
Some examples of composite numbers include 4, 6, 8, 9, and 12. These numbers have multiple factors which make them composite.
Divisibility
Divisibility refers to the characteristic of being divisible by a certain number without leaving a remainder. For example, 28 is divisible by 1, 2, 4, 7, 14, and 28. When determining if a number is prime or composite, understanding divisibility is crucial.

To check divisibility:
  • See if the number can be divided by smaller numbers evenly (i.e., no remainder).
  • List out all the divisors to help identify whether the number is prime or composite.
If a number can be divided evenly by numbers other than 1 and itself, it is composite. If it only divides evenly by 1 and itself, it is prime. For example, in the exercise, 28 is identified as a composite number because it has divisors other than just 1 and 28.

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