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Chapter 1: Problem 16

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

Short Answer

Expert verified
Neither

Step by step solution

01

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.
02

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.
03

Conclusion

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

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Key Concepts

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

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鈥攁nd also the only even鈥攑rime 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.

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Most popular questions from this chapter

Evaluate. $$ \frac{x^{2}+2^{x}}{x^{2}-2^{x}}, \text { for } x=3 $$

Simplify. Match the algebraic expression with the equivalent rewritten expression below. Check your answer by calculating the expression by hand and by using a calculator. A) \((5(3-7)+4 \wedge 3) /(-2-3)^{2}\) B) \((5(3-7)+4 \wedge 3) /\left(-2-3^{2}\right)\) C)\((5(3-7)+4) \wedge 3 /-2-3^{2}\) D) \(5(3-7)+4 \wedge 3 /(-2-3)^{2}\) $$ \frac{5(3-7)+4^{3}}{(-2-3)^{2}} $$

Simplify using a calculator. Round your answer to the nearest thousandth. $$ \frac{2.5^{2}-10 \cdot 12 \div(-1.5)}{(3+5)^{2}-60} $$

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Given that \(0.3 \overline{3}=\frac{1}{3}\) and \(0.6 \overline{6}=\frac{2}{3},\) express each of the following as a ratio of two integers. $$7.7 \overline{7}$$
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