Problem 37 - Basic College Mathematics

Question

Determine whether each number is prime, composite, or neither. $$ 1 $$

Step-by-Step Solution

Answer

The number 1 is neither prime nor composite.

Step 1: Understand the definitions

A prime number is a natural number greater than 1 that only has two distinct positive divisors: 1 and itself. A composite number is a natural number greater than 1 that has more than two distinct positive divisors. If a number is neither prime nor composite, it must be 1 or less.

Step 2: Analyze the number 1

Number 1 has only one positive divisor, which is itself. According to the definitions, a prime number needs to have exactly two distinct positive divisors, and a composite number needs to have more than two.

Step 3: Determine the classification

Since the number 1 does not meet the criteria for being a prime number or a composite number, it is classified as neither.

Key Concepts

Prime Numbers

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. This means it can only be divided evenly by 1 and the number itself. For example, the number 2 is prime because it can only be divided by 1 and 2 without leaving a remainder.
Here are some additional examples of prime numbers:

• 3
• 5
• 7
• 11
• 13

Prime numbers have exactly two distinct positive divisors. They play a key role in various fields of mathematics, especially in number theory. Recognizing and categorizing them is important for understanding the structure of numbers and their properties.

Composite Numbers

Composite numbers are natural numbers greater than 1 that have more than two distinct positive divisors. This means there are multiple ways to divide the number evenly beyond just 1 and the number itself. For instance, the number 4 is composite because it can be divided evenly by 1, 2, and 4.
Some examples of composite numbers are:

• 4
• 6
• 8
• 9
• 10

These numbers can be factored into smaller natural numbers, which means they have multiple factors. The ability to break down composite numbers into prime factors is a fundamental aspect of many mathematical processes, such as finding the greatest common divisor or simplifying fractions.

Number Classification

Number classification helps to categorize different numbers into defined groups based on their properties. The main categories include prime numbers, composite numbers, and a special case for the number 1.
Prime numbers are natural numbers greater than 1 with exactly two distinct positive divisors: 1 and themselves. On the other hand, composite numbers are natural numbers greater than 1 with more than two distinct positive divisors. The number 1 is unique because it doesn't fit into either category; it has only one positive divisor (itself), making it neither prime nor composite.
Understanding these distinctions is essential for various mathematical concepts and applications. It enhances problem-solving skills and helps in advanced topics such as cryptography and algorithm development. Knowing whether a number is prime, composite, or neither allows students to approach and solve problems more effectively.