GCF And LCM - Using The Ladder Method

🔑 Key Concepts

• The ladder method can be used to find both the greatest common factor and the least common multiple.
• The GCF is found by multiplying the common factors written on the side of the ladder.
• The LCM is found by multiplying the side factors and the remaining numbers at the bottom.
• If the numbers at the bottom still share a common factor, the ladder continues for another step.
• This method helps students organize factor relationships efficiently without making long factor lists.

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

Example 1 – GCF and LCM of 36 and 84

Step 1: Write the two numbers side by side.

\(36,\ 84\)

Step 2: Divide both numbers by common factors.

Divide both numbers by 2:

\(36\div2=18\)

\(84\div2=42\)

Divide both numbers by 6:

\(18\div6=3\)

\(42\div6=7\)

The remaining numbers, 3 and 7, have no common factor greater than 1.

Step 3: Find the GCF. Multiply the common factors used.

\(2\times6=12\)

Step 4: Find the LCM. Multiply the common factors and the remaining 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 the two numbers side by side.

\(42,\ 98\)

Step 2: Divide both numbers by common factors.

Divide both numbers by 2:

\(42\div2=21\)

\(98\div2=49\)

Divide both numbers by 7:

\(21\div7=3\)

\(49\div7=7\)

The remaining numbers, 3 and 7, have no common factor greater than 1.

Step 3: Find the GCF. Multiply the common factors used.

\(2\times7=14\)

Step 4: Find the LCM. Multiply the common factors and the remaining 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 the two numbers side by side.

\(36,\ 48\)

Step 2: Divide both numbers by common factors.

Divide both numbers by 6:

\(36\div6=6\)

\(48\div6=8\)

Divide both numbers by 2:

\(6\div2=3\)

\(8\div2=4\)

The remaining numbers, 3 and 4, have no common factor greater than 1.

Step 3: Find the GCF. Multiply the common factors used.

\(6\times2=12\)

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

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

Final Answers:

\(\operatorname{GCF}=12\)

\(\operatorname{LCM}=144\)

🧠 Math Vocabulary

• Factor: a whole number that divides another number evenly.
• Multiple: the product of a number and a whole number.
• Greatest Common Factor (GCF): the largest factor two numbers have in common.
• Least Common Multiple (LCM): the smallest multiple two numbers share.
• Ladder Method: an organized division strategy used to find both GCF and LCM.

💡 Main Idea

This video shows how the ladder method can be used to find both the greatest common factor and least common multiple in one organized setup. Students see that the side factors determine the GCF, while the side factors together with the remaining bottom numbers determine the LCM.

📚 What You Should Already Know

Students should already know basic multiplication facts, division facts, and how to recognize when one number divides evenly into another. It also helps to understand the difference between factors and multiples before using the ladder method.

🚀 What Comes Next

After students understand this method, they can apply it to simplifying fractions, finding common denominators, and solving multi-step number sense problems involving factors and multiples. This also prepares them for more advanced work with ratios, proportions, and rational numbers.

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