LCM and GCF — How to Find Least Common Multiple and Greatest Common Factor

Grade: 6–7 | Topic: Arithmetic

What You Will Learn

By the end of this page, you will be able to find the Greatest Common Factor (GCF) and Least Common Multiple (LCM) of two or more numbers using multiple methods. You will also understand when to use each one — GCF for simplifying fractions and LCM for adding fractions with unlike denominators.

Theory

What Is the GCF (Greatest Common Factor)?

The Greatest Common Factor of two numbers is the largest number that divides evenly into both of them.

Think of it as the biggest "shared divisor."

Example: For 12 and 18 — the factors of 12 are 1, 2, 3, 4, 6, 12. The factors of 18 are 1, 2, 3, 6, 9, 18. The common factors are 1, 2, 3, 6. The greatest one is 6, so GCF(12, 18) = 6.

Method 1 — List all factors:

1. Write out all factors of each number
2. Circle the ones that appear in both lists
3. The largest circled number is the GCF

Method 2 — Prime factorization:

1. Write each number as a product of prime factors
2. Identify the prime factors that appear in both
3. Multiply them together

$\text{GCF}(12, 18): \quad 12 = 2^2 \times 3, \quad 18 = 2 \times 3^2$

Both share one $2$ and one $3$, so $\text{GCF} = 2 \times 3 = 6$.

What Is the LCM (Least Common Multiple)?

The Least Common Multiple of two numbers is the smallest number that both numbers divide into evenly.

Think of it as the smallest "shared multiple."

Example: For 4 and 6 — multiples of 4: 4, 8, 12, 16, 20, 24... Multiples of 6: 6, 12, 18, 24... The first multiple that appears in both lists is 12, so LCM(4, 6) = 12.

Method 1 — List multiples:

1. Write out multiples of each number
2. Find the first multiple that appears in both lists

Method 2 — Prime factorization:

1. Write each number as a product of prime factors
2. Take the highest power of every prime factor that appears in either number
3. Multiply them together

$\text{LCM}(4, 6): \quad 4 = 2^2, \quad 6 = 2 \times 3$

Take the highest power of each prime: $2^2$ and $3^1$, so $\text{LCM} = 4 \times 3 = 12$.

When to Use GCF vs LCM

Situation	Use
Simplify a fraction $\frac{12}{18}$	GCF — divide top and bottom by GCF(12, 18) = 6
Add $\frac{1}{4} + \frac{1}{6}$	LCM — use LCM(4, 6) = 12 as common denominator
Split items into equal groups as large as possible	GCF
Find when two events repeat at the same time	LCM

Worked Examples

Example 1: Finding GCF by Listing Factors

Problem: Find GCF(24, 36).

Step 1: List all factors of 24. $24: \quad 1, 2, 3, 4, 6, 8, 12, 24$

Step 2: List all factors of 36. $36: \quad 1, 2, 3, 4, 6, 9, 12, 18, 36$

Step 3: Identify the common factors: 1, 2, 3, 4, 6, 12.

Answer: GCF(24, 36) = 12

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Example 2: Finding LCM Using Prime Factorization

Problem: Find LCM(8, 12).

Step 1: Write the prime factorization of each number. $8 = 2^3 \qquad 12 = 2^2 \times 3$

Step 2: Take the highest power of every prime that appears.

• Highest power of 2: $2^3$
• Highest power of 3: $3^1$

Step 3: Multiply them. $\text{LCM} = 2^3 \times 3 = 8 \times 3 = 24$

Answer: LCM(8, 12) = 24

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Example 3: Applying GCF to Simplify a Fraction

Problem: Simplify $\dfrac{30}{45}$ to lowest terms.

Step 1: Find GCF(30, 45). $30 = 2 \times 3 \times 5 \qquad 45 = 3^2 \times 5$ Common factors: $3 \times 5 = 15$, so GCF = 15.

Step 2: Divide numerator and denominator by 15. $\frac{30 \div 15}{45 \div 15} = \frac{2}{3}$

Answer: $\dfrac{30}{45} = \mathbf{\dfrac{2}{3}}$

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Example 4: Applying LCM to Add Fractions

Problem: Calculate $\dfrac{5}{6} + \dfrac{3}{8}$.

Step 1: Find LCM(6, 8). $6 = 2 \times 3 \qquad 8 = 2^3$ $\text{LCM} = 2^3 \times 3 = 24$

Step 2: Convert both fractions to have denominator 24. $\frac{5}{6} = \frac{5 \times 4}{24} = \frac{20}{24} \qquad \frac{3}{8} = \frac{3 \times 3}{24} = \frac{9}{24}$

Step 3: Add. $\frac{20}{24} + \frac{9}{24} = \frac{29}{24}$

Answer: $\mathbf{\dfrac{29}{24}}$ (or $1\dfrac{5}{24}$)

Common Mistakes

Mistake 1: Confusing GCF and LCM

❌ To add $\frac{1}{4} + \frac{1}{6}$, student uses GCF(4, 6) = 2 as the denominator.

✅ Use LCM(4, 6) = 12 as the common denominator: $\frac{3}{12} + \frac{2}{12} = \frac{5}{12}$.

Why this matters: GCF divides numbers down. LCM builds up to a common ground. These are opposite operations applied to different problems.

Mistake 2: Stopping at a Common Multiple that is Not the Least

❌ LCM(4, 6) = 24 (correct that it's common, but 12 is smaller).

✅ Always check for smaller common multiples before accepting the first one you find. List multiples systematically.

Mistake 3: Missing Prime Factors in Factorization

❌ $36 = 4 \times 9$ (not a prime factorization — 4 and 9 are composite).

✅ $36 = 2^2 \times 3^2$ (keep breaking down until all factors are prime).

Practice Problems

Try these on your own before checking the answers:

1. Find GCF(16, 24).
2. Find LCM(5, 9).
3. Simplify $\dfrac{42}{56}$ using the GCF.
4. Add $\dfrac{2}{9} + \dfrac{1}{6}$ using the LCM.
5. Two buses leave a station at the same time. Bus A comes every 12 minutes; Bus B every 8 minutes. How many minutes until they leave together again?

Click to see answers

1. $16 = 2^4$, $24 = 2^3 \times 3$. GCF = $2^3 = \mathbf{8}$
2. $5 = 5$, $9 = 3^2$. No shared primes, so LCM = $5 \times 9 = \mathbf{45}$
3. GCF(42, 56) = 14. $\frac{42 \div 14}{56 \div 14} = \mathbf{\dfrac{3}{4}}$
4. LCM(9, 6) = 18. $\frac{4}{18} + \frac{3}{18} = \mathbf{\dfrac{7}{18}}$
5. LCM(12, 8) = 24. They leave together again in 24 minutes.

Summary

• GCF = largest number dividing evenly into both numbers. Use prime factorization: multiply the shared prime factors (lowest power).
• LCM = smallest number both numbers divide into. Use prime factorization: multiply all prime factors (highest power).
• Use GCF to simplify fractions; use LCM to find a common denominator.
• When two numbers share no prime factors, GCF = 1 and LCM = their product.

Related Topics

• Fractions — Complete Guide
• How to Add Fractions with Unlike Denominators
• How to Simplify Fractions

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