Problem 16 - Elementary and Intermediate Algebra

Question

Label each of the following numbers as prime, composite, or neither. $$1$$

Step-by-Step Solution

Answer

Neither

Step 1: Understand the Definitions

Define the terms prime, composite, and neither. 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 more than two positive divisors. 'Neither' refers to numbers that do not fit into the prime or composite categories.

Step 2: Evaluate the Number 1

Check if the number 1 is greater than 1. The number 1 is not greater than 1, so it does not fit the definitions of either prime or composite numbers.

Step 3: Conclusion

Since 1 does not meet the criteria to be classified as a prime or composite number, label it as 'neither'.

Key Concepts

Prime Numbers

Prime numbers are special types of numbers that you will encounter often in mathematics. A prime number is defined as a natural number greater than 1, which has no positive divisors other than 1 and itself.

For example, consider the number 7:

• It is greater than 1.
• The only divisors of 7 are 1 and 7.

Because it meets both criteria, 7 is a prime number.
An easy way to remember a prime number is to think of it as a 'building block' of other numbers. These cannot be broken down or divided into smaller integers without leaving a remainder.

Another example is the number 2, which is the smallest—and also the only even—prime number.

Composite Numbers

Composite numbers are opposite sides of the coin when compared to prime numbers. A composite number is any natural number greater than 1 that has more than two positive divisors.

Take the number 4 for instance:

• It is greater than 1.
• It can be divided by 1, 2, and 4.

Since 4 has more than two divisors, it is considered a composite number.

In simple terms, composite numbers can be 'broken down' into simpler components. For instance, 4 can be written as 2 multiplied by 2.
Understanding composite numbers is crucial, as it helps in factorization and finding common divisors in mathematics.

Another example is 6 which has divisors 1, 2, 3, and 6 itself.

Neither Category

Not all numbers fit neatly into the prime or composite categories. There are numbers that do not satisfy the criteria for either.

The number 1 is a prime example of this:

• It is not greater than 1.
• It doesn't fit the definition of a prime number since a prime number must be greater than 1.
• It can't be composite because it doesn't have more than two divisors.

Consequently, the number 1 is classified as 'neither'.

Understanding what numbers fall into the 'neither' category helps prevent confusion. It's important to recognize that while 1 is a unique number, some numbers are simply too special to fit into general categories, including 0 and sometimes negative numbers in specific contexts.