Prime Factorization As A Way To Find The GCF And LCM

🔑 Key Concepts

• A prime number has exactly two factors: \(1\) and itself.
• A composite number has more than two factors.
• A factor tree breaks a number into prime factors.
• The GCF can be found by identifying the prime factors the numbers have in common.
• The LCM can be found by taking all prime factors needed to build both numbers.

✏️ Worked Examples – Using Factor Trees to Find GCF and LCM

Example 1 – Find the GCF Using Factor Trees

Find the GCF of 24 and 18.

Step 1: Create a factor tree for each number. Continue breaking apart composite factors until every branch ends with a prime number.

Prime factorization of 24:

\(\displaystyle 24=2\times2\times2\times3\)

Prime factorization of 18:

\(\displaystyle 18=2\times3\times3\)

Step 2: Match the prime factors that appear in both factorizations.

Matching common prime factors:

\(\displaystyle 2\text{ and }3\)

Step 3: Multiply the matching common prime factors.

\(\displaystyle 2\times3=6\)

Final Answer:

\(\boxed{\operatorname{GCF}(24,18)=6}\)

Example 2 – Find the LCM Using Factor Trees

Find the LCM of 24 and 18.

Step 1: Use the factor trees to write the prime factorization of each number.

Prime factorization of 24:

\(\displaystyle 24=2\times2\times2\times3\)

Prime factorization of 18:

\(\displaystyle 18=2\times3\times3\)

Step 2: Match the common prime factors between the two trees.

\(\displaystyle 2\times3=6\)

Step 3: Include the unmatched prime factors needed to represent both numbers.

Prime factors used for the LCM:

\(\displaystyle 2\times2\times2\times3\times3\)

Step 4: Multiply all the selected prime factors.

\(\displaystyle 2\times2\times2\times3\times3=72\)

Another way to use the diagram is to multiply the unmatched factor of 3 by 24.

\(\displaystyle 3\times24=72\)

Final Answer:

\(\boxed{\operatorname{LCM}(24,18)=72}\)

🧠 Math Vocabulary

• Prime number: a number with exactly two factors, \(1\) and itself.
• Composite number: a number with more than two factors.
• Prime factorization: writing a number as a product of prime numbers.
• Factor tree: a diagram used to break a number into its prime factors.
• Common prime factors: prime factors shared by two numbers, useful for finding the GCF.

💡 Main Idea

This lesson connects prime numbers, composite numbers, and factor trees to two important ideas: greatest common factor and least common multiple. By breaking numbers into prime factors, students can see the structure of each number more clearly and use that structure to compare numbers efficiently.

📚 What You Should Already Know

Before watching this video, students should understand factors, multiples, and basic multiplication facts. It also helps if they already know the difference between a prime number and a composite number.

🚀 What Comes Next

After learning how to use factor trees, students can apply prime factorization to simplify fractions, find GCF and LCM more efficiently, and solve number theory problems involving divisibility and equivalent forms.

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