Finding The GCF And LCM Using the Ladder Method

🔑 Key Concepts

• A factor divides a number evenly with no remainder.
• The greatest common factor (GCF) is the largest factor shared by two numbers.
• A multiple is found by counting by a number or multiplying it by whole numbers.
• The least common multiple (LCM) is the smallest number that appears in both multiple lists.
• The ladder method can be used to find both the GCF and the LCM efficiently.

✏️ Worked Examples – Using the Ladder Method to Find GCF and LCM

Example 1 – GCF and LCM of 36 and 84

Step 1: Write both numbers side by side.

\(36,\ 84\)

Step 2: Divide both numbers by common factors until the remaining numbers have no common factor greater than 1.

Common factors used:

\(2\qquad 6\)

Step 3: Find the GCF. Multiply the common factors on the left side of the ladder.

\(2\times6=12\)

Step 4: Find the LCM. Multiply the common factors and the remaining bottom numbers.

\(2\times6\times3\times7=252\)

Final Answers:

\(\operatorname{GCF}=12\)

\(\operatorname{LCM}=252\)

Example 2 – GCF and LCM of 42 and 98

Step 1: Write both numbers side by side.

\(42,\ 98\)

Step 2: Divide both numbers by common factors until the ladder is complete.

Common factors used:

\(2\qquad 7\)

Step 3: Find the GCF. Multiply the common factors on the left.

\(2\times7=14\)

Step 4: Find the LCM. Multiply the common factors and the remaining bottom numbers.

\(2\times7\times3\times7=294\)

Final Answers:

\(\operatorname{GCF}=14\)

\(\operatorname{LCM}=294\)

Example 3 – GCF and LCM of 36 and 48

Step 1: Write both numbers side by side.

\(36,\ 48\)

Step 2: Divide both numbers by common factors until the remaining numbers share no factor greater than 1.

Common factors used:

\(6\qquad 2\)

Step 3: Find the GCF. Multiply the common factors on the left.

\(6\times2=12\)

Step 4: Find the LCM. Multiply the common factors and the remaining bottom numbers.

\(6\times2\times3\times4=144\)

Final Answers:

\(\operatorname{GCF}=12\)

\(\operatorname{LCM}=144\)

🧠 Math Vocabulary

• Factor: a number that divides another number evenly.
• Multiple: the result of multiplying a number by a whole number.
• Greatest Common Factor (GCF): the largest factor shared by two numbers.
• Least Common Multiple (LCM): the smallest multiple shared by two numbers.
• Ladder Method: a step-by-step division method used to find both the GCF and LCM.

💡 Main Idea

This lesson compares the greatest common factor and least common multiple using both factor lists and the ladder method. Students learn how to identify common factors, determine the GCF, and use that information to find the LCM efficiently. These skills are essential for simplifying fractions, solving ratio problems, and working with common denominators.

📚 What You Should Already Know

Students should understand multiplication facts, division facts, and how to determine whether one number divides evenly into another. Familiarity with factor pairs and multiplication tables will make it easier to identify common factors quickly.

🚀 What Comes Next

After learning to find the GCF and LCM using lists and the ladder method, students will apply these skills to simplifying fractions, finding common denominators, solving ratio problems, and working with rational numbers in more advanced algebra topics.

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