Problem 13 - Introductory and Intermediate Algebra for College Students 4th

Question

Identify each natural number as prime or composite. If the number is composite, find its prime factorization. $$22$$

Step-by-Step Solution

Answer

The number 22 is a composite number and its prime factorization is \(2 \times 11\).

Step 1: Identify the Type of the Number

First, determine if 22 is a prime or composite number. To do this, attempt dividing it by numbers other than 1 and itself. If it can be divided without a remainder by a number other than 1 and itself, it is a composite number. If not, it's a prime number. In this case, 22 divided by 2 gives a whole number (11), so 22 is a composite number.

Step 2: Find the Prime Factorization

Once it's been confirmed that 22 is a composite number, move onto finding its prime factorization. This process breaks down a composite number into its prime factors. To find the prime factorization of 22, start dividing it by the smallest prime number (2) and continue the process with the quotient until it's not possible. Here, 22 divided by 2 gives 11, which is a prime number and cannot be divided further. Therefore, the prime factorization of 22 is \(2 \times 11\).

Key Concepts

Natural Numbers

Natural numbers are a critical foundation in mathematics. They are the numbers we use for counting and ordering.
Starting from 1, these numbers include 2, 3, 4, and so on. Natural numbers do not include fractions, decimals, or negative numbers.
They are the simplest form of numbers and are often used in everyday counting and ordering tasks.

• **Origin:** Natural numbers are inherently positive and begin from 1, continuing infinitely.
• **Notation:** Represented typically by the symbol \( \mathbb{N} \).

Understanding natural numbers is essential as they form the basis for more complex mathematical concepts, including those of prime and composite numbers.

Composite Number

Composite numbers play an important role in the building blocks of mathematics. A composite number is a natural number greater than 1 that is not prime. Simply put, it has more than two factors.
If the number can be divided evenly by numbers other than 1 and itself, it is composite. For example, the number 22 is composite because it has factors other than 1 and itself – it can also be divided by 2 and 11.
This means:

• **More Than Two Factors:** Unlike prime numbers, composite numbers have multiple factors.
• **Prime Factorization:** Every composite number can be expressed as a product of prime numbers.

In the step-by-step solution, the number 22 was determined as composite, and its prime factorization is given by \(2 \times 11\). Recognizing composite numbers and their factorizations aids in understanding the structure of numbers.

Prime Number

Prime numbers are fascinating and hold a special place in mathematics. A prime number is a natural number greater than 1 that has no divisors other than 1 and itself.
This means it can only be divided evenly by 1 and the number itself. Prime numbers are like the "atoms" of the number world because other numbers are built from them.

• **Two Factors Only:** By definition, prime numbers have exactly two distinct positive divisors.
• **Examples:** Some early prime numbers include 2, 3, 5, 7, and 11.

The step-by-step solution to the exercise showed that 11 is a prime number since it cannot be divided further into other numbers without leaving a remainder. Understanding prime numbers is key to mastering number theory and factorization.