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Chapter 5: Problem 1

Write out the answers to these questions. Use complete sentences. What is a prime number?

Short Answer

Expert verified
A prime number is a natural number greater than 1 with exactly two positive divisors: 1 and itself.

Step by step solution

01

Define a Prime Number

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.
02

Properties of Prime Numbers

Prime numbers have exactly two distinct positive divisors: 1 and the number itself. This means that they cannot be formed by multiplying two smaller natural numbers.
03

Example of Prime Numbers

Examples of prime numbers include 2, 3, 5, 7, 11, 13, 17, 19, and so on. The number 2 is the only even prime number.
04

Non-Examples of Prime Numbers

Non-prime numbers, also known as composite numbers, include 4, 6, 8, 9, and 10. For instance, 4 can be divided by 1, 2, and 4.

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Key Concepts

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

natural numbers
Natural numbers are a set of positive integers starting from 1 and going upwards indefinitely. They are denoted by the symbol \( \mathbb{N} \ \). These numbers include: \(1, 2, 3, 4, 5, \ldots\ \ \). Natural numbers are often used for counting (like 1 apple, 2 apples) and ordering (1st, 2nd, 3rd, etc.). Natural numbers do not include zero or any negative numbers.
To understand prime numbers, it's important first to grasp what natural numbers are because prime numbers are a specific type of natural number.
Here are some properties of natural numbers:
  • Every natural number is greater than 0.
  • Natural numbers form the basis of all other number sets used in mathematics, including whole numbers, integers, rational numbers, irrational numbers, and real numbers.
number theory
Number theory is a branch of mathematics that deals with properties and relationships of numbers, particularly integers. It's often referred to as the 'queen of mathematics' because of its deep and fundamental nature.
Number theory involves various concepts such as prime numbers, composite numbers, divisibility, and the greatest common divisor.
One core area in number theory is the study of prime numbers and their distribution. Mathematicians are particularly interested in understanding which numbers are prime and how often they occur.
Some important concepts in number theory related to prime numbers include:
  • Prime Factorization: This is the representation of a composite number as the product of its prime factors. For example, 12 can be factorized to \(2^2 \times 3\ \ \).
  • The Fundamental Theorem of Arithmetic: This states that every integer greater than 1 can be uniquely represented as a product of prime numbers.
composite numbers
Composite numbers are natural numbers that have more than two distinct positive divisors. Unlike prime numbers that have no divisors other than 1 and themselves, composite numbers can be divided by at least one number besides 1 and their own value.
For example, take the number 4. Besides being divisible by 1 and 4, it can also be divided evenly by 2. Therefore, 4 is a composite number.
Some common examples of composite numbers include 4, 6, 8, 9, 10, 12, and 14. On the other hand, prime numbers like 2, 3, 5, 7, and 11 cannot be divided by any other number except for 1 and themselves.
Here are a few key points on composite numbers:
  • A composite number is always greater than 1.
  • Every composite number can be expressed as a product of two or more prime numbers. For example, 12 can be written as 2 \(\times\ \) 6 or 3 \(\times\ \) 4.
  • All even numbers greater than 2 are composite numbers because they are divisible by 2.

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Most popular questions from this chapter

Factor each polynomial using the trial-and-error method. $$ a^{2}-8 a-33 $$

Factor each polynomial completely. $$ 8 x^{2}-2 x-15 $$

Factor each polynomial by grouping. $$ x t-t^{2}-a x+a t $$

Factor each polynomial using the trial-and-error method. $$ p^{2}+9 p+20 $$

Factor each polynomial and explain how you decided which method to use. a) \(x^{2}+10 x+25\) b) \(x^{2}-10 x+25\) c) \(x^{2}+26 x+25\) d) \(x^{2}-25\) e) \(x^{2}+25\)
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