Problem 102 - Algebra 1

Question

Write the prime factorization. (Skills Review, p. \(T T T\) ) $$24$$

Step-by-Step Solution

Answer

The prime factorization of 24 is \(2^3 * 3\).

Step 1: Find the First Pair of Factors

First, identify a pair of factors for 24. You can start with the number 1 and the number itself (24). However, since 24 isn't a prime number, this step is typically bypassed in favor of finding two factors where one of them is a prime number. A pair of factors of 24 include 2 and 12, so you can write 24 as 2*12.

Step 2: Break Down Composite Factors

Continuing the factor tree, identify factors of 12. Since 12 isn't a prime number, find a pair of factors where one of them is a prime number. A pair of factors could be 2 and 6. Write 12 as 2*6 to continue the factor tree.

Step 3: Continue Until All Factors are Prime

The number 6 is a composite number, and it can be written as a product of two primes, 2 and 3. Now, the number 2 and 3 are prime numbers and cannot be factored any further.

Step 4: Write the Prime Factorization

Put all the prime numbers you identified together to write the prime factorization of 24. So, 24 can be written as the product of these prime numbers: \(2^3 * 3\).

Key Concepts

Factors

Factors are the numbers you multiply together to obtain another number, known as the product. Understanding factors is a primary concept in algebra and arithmetic which students often encounter early in their math studies. Consider it like finding the building blocks of a number.

For instance, if you have the number 24, factors are the pairs of numbers you multiply together to get 24. Some examples of factor pairs for 24 would be:

• 1 and 24
• 2 and 12
• 3 and 8
• 4 and 6

Each of these pairs, when multiplied together, results in 24. It's important to remember that factors of a number can include the number itself and 1, as every number is divisible by these two numbers. Finding factors becomes crucial when you move into prime factorization, as it helps you break down numbers into their simplest prime components.

Composite Numbers

Composite numbers are numbers with more than two distinct positive divisors. Simply put, if a number can be divided evenly by numbers other than just 1 and itself, it falls into the category of composite numbers.

For example, the number 24 is a classic composite number. It has multiple factors: 1, 2, 3, 4, 6, 8, 12, and 24. Because of this, it can be broken down into a product of other numbers, unlike a prime number which can only be divided by 1 and itself.

• On its factor tree, you start by selecting any convenient pair of factors like 2 and 12.
• Further examine 12 by dividing it into 2 and 6.
• Look at 6, which can be split into yet another factor pair of 2 and 3, both of which are prime.

This process of breaking down composite numbers continues until all the remaining factors are prime numbers.

Prime Numbers

Prime numbers are the fundamental particles in the realm of numbers, as they can't be divided by anything other than 1 and themselves. They are not only the building blocks in prime factorization, but also one of the simplest and most crucial groups of numbers to understand.

Examples of prime numbers include 2, 3, 5, 7, 11, and so forth. These are numbers that you can't split into any other divisors, making them completely unique in multiplication. When you're breaking down a composite number, uncovering these prime numbers is the ultimate goal.

• In our scenario of 24, the goal is to break it down using smaller and smaller numbers until you're left with 2s and 3.
• This process gives us a multi-factor product: \(2^3 \times 3\).

By understanding and finding prime numbers, you can simplify and solve various mathematical problems more efficiently.