Problem 10 - Prealgebra

Question

Factor each of the following into a product of prime factors. $$8$$

Step-by-Step Solution

Answer

The prime factorization of 8 is \(2^3\).

Step 1: Identify if the number is prime

Check if the number 8 is a prime number. A prime number has exactly two distinct positive divisors: 1 and itself. Since 8 can be divided by 1, 2, 4, and 8, it is not a prime number.

Step 2: Find the smallest prime factor

Start by dividing the number 8 by the smallest prime number, which is 2. Since 8 is even, it can be divided by 2. Perform the division: \[ 8 \div 2 = 4 \]

Step 3: Continue factoring

Now, divide the result (4) by the smallest prime factor, which is again 2. \[ 4 \div 2 = 2 \]

Step 4: Reach the prime number

Now, notice the result is a prime number (2). So we can stop dividing further.

Step 5: Write the complete factorization

Putting it all together, we express 8 as a product of its prime factors: \[ 8 = 2 \times 2 \times 2 \] Or simply, \[ 8 = 2^3 \]

Key Concepts

Understanding Prime Numbers

Prime numbers are the building blocks of all numbers. They are numbers greater than 1 and can only be divided evenly by 1 and themselves. This means that they have exactly two distinct positive divisors. The sequence of prime numbers starts with 2, followed by 3, 5, 7, and so on. It is important to note that the number 2 is the smallest and the only even prime number.
All other prime numbers are odd because any even number greater than 2 can be divided by 2 and, hence, cannot be prime. Recognizing prime numbers is crucial when it comes to factoring, as these numbers will form the core of any number's decomposition into factors.

Factoring Numbers

Factoring numbers involves breaking down a number into a product of smaller numbers, specifically prime numbers, to understand its basic components. This means expressing the original number as a multiplication of these prime factors. For example, when we factor the number 8, we express it as the product of three 2's.

This means:

• The number is decomposed entirely into products of prime numbers.
• This highlights the unique prime factors that make up the original number.

The process of factoring is used in various areas of mathematics, including simplifying fractions, finding greatest common divisors, and solving equations. Factoring forms a fundamental skill in algebra and serves as a gateway to understanding complex mathematical relationships.

Division Method for Prime Factorization

The division method for prime factorization is a systematic approach to break down any given number into its prime factors. It involves dividing the number by the smallest possible prime number and continuing to divide the quotient by prime numbers until we reach 1. Let's explore the method:

• Start with the smallest prime number, which is 2.
• If the number is divisible by 2, divide it and continue the process with the quotient.
• If the quotient is an even number, it can still be divided by 2.
• When reaching an odd quotient, check divisibility by the next smallest prime numbers such as 3, 5, and so on.

The process stops when the final quotient is 1, and the factors used during division are the prime factors of the original number. An example is factoring 8 as repeated divisions by 2 until reaching 1, resulting in the factors 2 x 2 x 2 or \(2^3\). This method is efficient, especially for larger numbers, as it systematically tests divisibility by prime numbers.