Problem 6 - Prealgebra

Question

Find the GCF of each set of numbers. $$6,15,24$$

Step-by-Step Solution

Answer

The GCF of 6, 15, and 24 is 3.

Step 1: Prime Factorization

List the prime factors of each number. - **6**: The prime factors of 6 are 2 and 3. So, 6 = 2 × 3. - **15**: The prime factors of 15 are 3 and 5. So, 15 = 3 × 5. - **24**: The prime factors of 24 are 2, 2, 2, and 3. So, 24 = 2^3 × 3.

Step 2: Identify Common Prime Factors

Determine which prime factors appear in all the numbers. - Common prime factors: The number 3 appears in the prime factorization of all three numbers: 6 = 2 × 3, 15 = 3 × 5, and 24 = 2^3 × 3.

Step 3: Find the Greatest Common Factor

Calculate the greatest common factor by taking the lowest power of the common prime factor(s) found in all the numbers. - Here, the common prime factor is 3, which appears in each list of prime factors, and is not raised to any higher power. Thus, the GCF is simply 3.

Key Concepts

Prime Factorization

Prime factorization is the process of breaking down a number into its most basic building blocks—prime numbers. Prime numbers are numbers greater than 1 that can only be divided by 1 and themselves without leaving a remainder. For example, the prime factors of 6 are 2 and 3 because when multiplied together, they give 6 back: 6 = 2 × 3.
When performing prime factorization, you repeatedly divide the number by the smallest possible prime until you're left with only prime numbers. It's like peeling back the layers of an onion until you reach the core. For 24, the process goes like this:

• 24 divided by 2 equals 12.
• 12 divided by 2 equals 6.
• 6 divided by 2 equals 3.
• 3 is already a prime number.

Thus, the prime factorization of 24 is 2^3 × 3.* Practice makes perfect! The more you practice this process, the easier it becomes.

Common Prime Factors

The step of identifying common prime factors is crucial in determining relationships between numbers. After finding the prime factors, the next task is to seek out which prime factors the numbers share.
Taking the numbers 6, 15, and 24:

• 6 is broken down into 2 and 3.
• 15 breaks down into 3 and 5.
• 24 breaks down into 2, 2, 2, and 3.

Scanning each prime factorization, we see that the number 3 appears in the factorization of each number. In this step, you aren't looking for quantity (how many times the number appears), you're just confirming presence. This concept highlights a connection between the numbers. The presence of 3 in each set demonstrates a commonality, which helps when calculating results like the greatest common factor.

Integer Factorization

Integer factorization involves breaking down an integer (a whole number) into other integers that multiply together to give the original number. This process isn't limited to just prime numbers, unlike prime factorization, and can reveal relationships between different integers.
Consider the integer factorization of 24. It can be:

• 2 × 12
• 3 × 8
• 4 × 6

Each pair multiplies to return 24. However, these are not the prime factors; they're just regular factors. To focus solely on integer factorization, think of it as finding the basic building blocks of numbers via multiplication.
This method assists in understanding numbers more deeply and can be the first step before narrowing down to prime factorization. Once integer factorization is grasped, identifying prime factors or greatest common factors becomes a smooth and logical progression.