Problem 47 - Basic College Mathematics

Question

Find the prime factorization of each number. $$ 14 $$

Step-by-Step Solution

Answer

The prime factorization of 14 is 2 × 7.

Step 1: Identify the smallest prime number

The smallest prime number is 2. Check if 14 is divisible by 2.

Step 2: Divide the number by the smallest prime

Since 14 is divisible by 2, divide 14 by 2: \[ 14 ÷ 2 = 7 \]

Step 3: Determine if the quotient is a prime number

The quotient obtained, 7, is a prime number.

Step 4: Write the prime factorization

Combine the prime factors. Therefore, the prime factorization of 14 is: \[ 14 = 2 × 7 \]

Key Concepts

Prime Factorization Introduction

Let's dive into the concept of prime factorization using the example of finding the factors of the number 14. Prime factorization is the process of expressing a number as the product of prime numbers. It's a fundamental concept in mathematics that helps simplify problems and understand the structure of numbers. In this guide, we'll walk through the concepts needed to fully understand prime factorization, including prime numbers, division, and mathematical steps.

Prime Numbers

Prime numbers play a key role in prime factorization. A prime number is a number greater than 1 that has no divisors other than 1 and itself.
Some examples of prime numbers include:

• 2
• 3
• 5
• 7
• 11

For instance, the number 7 is a prime number because it can't be divided evenly by any other numbers except 1 and 7.
In our exercise, we identified 2 as the smallest prime number. Knowing prime numbers and being able to identify them quickly makes the process of prime factorization easier and more efficient.

The Division Process

Division is a crucial step in prime factorization. This method involves dividing the given number by the smallest prime number until you're left with a quotient that is also a prime number. Let's go through the steps:
First, we check if 14 is divisible by 2 since 2 is the smallest prime number. Since \( 14 ÷ 2 = 7 \), we move on to the next step. Now we have the quotient, 7, which is indeed a prime number. This tells us that we have reached the end of the division process.
These steps are repeated for larger numbers, using increasingly larger primes as needed. The result is a list of prime factors that, when multiplied together, give you the original number.

Mathematical Steps for Prime Factorization

Understanding and following the correct mathematical steps is essential for prime factorization. Here's a detailed breakdown:

• Step 1: Identify the smallest prime number that can divide the given number.
• Step 2: Divide the original number by this prime number.
• Step 3: Look at the quotient from step 2. Check if this quotient is a prime number.
• Step 4: If the quotient is prime, list it as one of the factors. If it's not, repeat steps 1-3 using the quotient until a prime is obtained.
• Step 5: Write out the original number as the product of all the prime factors found.

For example, for 14, we did the following:

• \textbf{Step 1}: Identified 2 as the smallest prime.
• \textbf{Step 2}: Divided 14 by 2 to get 7.
• \textbf{Step 3}: Recognized that 7 is a prime number.
• \textbf{Step 4}: Concluded the factorization as 2 and 7.
• \textbf{Step 5}: Final solution is \(14 = 2 × 7\).

By following these steps, you can break down any integer into its prime factors, making it easier to handle complex mathematical problems.