Problem 133 - Introductory Algebra for College Students

Question

Describe the difference between a prime number and a composite number.

Step-by-Step Solution

Answer

Prime numbers are natural numbers greater than 1 with only two positive divisors, 1 and the number itself, hence they have only two distinct factors. Composite numbers are positive integers that have more than two positive divisors or factors. Prime numbers are often odd numbers (except 2), while composite numbers can be either even or odd.

Step 1: Understanding Prime Numbers

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. The smallest prime number is 2. Other examples of prime numbers include 3, 5, 7, 11 and 13. An important note is that by definition, 1 is not considered a prime number.

Step 2: Understanding Composite Numbers

A composite number is a positive integer that has at least one positive divisor other than one or itself. In other words, a composite number is any positive integer greater than one that is not a prime number. For example, 4, 6, 8, 9, and 10 are the first few composite numbers.

Step 3: Comparing Prime and Composite Numbers

In comparison, while prime numbers only have two distinct positive divisors, 1 and the number itself, composite numbers have more than two positive divisors. One key distinction between prime and composite numbers is that all prime numbers are odd (except 2), whereas composite numbers can be either even or odd.