GCF and LCM

What is GCF?

GCF stands for Greatest Common Factor

It's the largest number that divides evenly into two or more numbers.

Example: GCF of 12 and 18

Factors of 12: 1, 2, 3, 4, 6, 12 Factors of 18: 1, 2, 3, 6, 9, 18

Common factors: 1, 2, 3, 6

Greatest Common Factor (GCF) = 6 ✓

6 is the biggest number that divides evenly into both 12 and 18!

Finding GCF - List Method

Step 1: List all factors of each number Step 2: Find the common factors Step 3: Pick the greatest (biggest) one

Example: GCF of 24 and 36

Step 1: List factors

• Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
• Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36

Step 2: Common factors 1, 2, 3, 4, 6, 12

Step 3: Greatest common factor GCF = 12 ✓

Finding GCF - Prime Factorization Method

Step 1: Find prime factorization of each number Step 2: Circle the common prime factors Step 3: Multiply the common primes together

Example: GCF of 30 and 45

Step 1: Prime factorization

• 30 = 2 × 3 × 5
• 45 = 3 × 3 × 5

Step 2: Common primes

• Both have: 3 and 5

Step 3: Multiply common primes GCF = 3 × 5 = 15 ✓

What is LCM?

LCM stands for Least Common Multiple

It's the smallest number that is a multiple of two or more numbers.

Example: LCM of 4 and 6

Multiples of 4: 4, 8, 12, 16, 20, 24, 28... Multiples of 6: 6, 12, 18, 24, 30...

Common multiples: 12, 24, 36...

Least Common Multiple (LCM) = 12 ✓

12 is the smallest number that both 4 and 6 divide into evenly!

Finding LCM - List Method

Step 1: List multiples of each number Step 2: Find the common multiples Step 3: Pick the least (smallest) one

Example: LCM of 3 and 5

Step 1: List multiples

• Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30...
• Multiples of 5: 5, 10, 15, 20, 25, 30...

Step 2: Common multiples 15, 30, 45, 60...

Step 3: Least common multiple LCM = 15 ✓

Finding LCM - Prime Factorization Method

Step 1: Find prime factorization of each number Step 2: Take each prime factor the MOST times it appears Step 3: Multiply them together

Example: LCM of 12 and 18

Step 1: Prime factorization

• 12 = 2 × 2 × 3
• 18 = 2 × 3 × 3

Step 2: Take each prime the most it appears

• 2 appears twice in 12 → take 2²
• 3 appears twice in 18 → take 3²

Step 3: Multiply LCM = 2² × 3² = 4 × 9 = 36 ✓

GCF vs. LCM

GCF (Greatest Common Factor):

• Find the BIGGEST number that goes into both
• Smaller than both numbers (or equal)
• Use for: simplifying fractions, dividing into groups

LCM (Least Common Multiple):

• Find the SMALLEST number both go into
• Bigger than both numbers (or equal)
• Use for: adding/subtracting fractions, finding patterns

Memory trick:

• GCF = Goes into (divides into)
• LCM = Larger number (usually)

Real-World Examples

GCF Example: Sharing Equally

Problem: You have 24 cookies and 36 brownies. What's the largest number of identical gift bags you can make (using all items)?

Solution: Find GCF of 24 and 36

Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24 Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36

GCF = 12

Answer: You can make 12 gift bags! ✓

• Each bag has 2 cookies (24 ÷ 12 = 2)
• Each bag has 3 brownies (36 ÷ 12 = 3)

LCM Example: Events Repeating

Problem: The pizza shop delivers every 3 days. The sushi shop delivers every 4 days. If both deliver today, when will they both deliver on the same day again?

Solution: Find LCM of 3 and 4

Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24... Multiples of 4: 4, 8, 12, 16, 20, 24...

LCM = 12

Answer: In 12 days! ✓

Special Cases

When One Number is a Multiple of the Other

Example: GCF and LCM of 6 and 12

6 goes into 12 evenly!

GCF = 6 (the smaller number) LCM = 12 (the larger number)

Rule: When one number divides evenly into another:

• GCF = smaller number
• LCM = larger number

When Numbers Have No Common Factors (Relatively Prime)

Example: GCF and LCM of 8 and 9

Factors of 8: 1, 2, 4, 8 Factors of 9: 1, 3, 9

Only common factor is 1!

GCF = 1 LCM = 8 × 9 = 72

Rule: When GCF = 1:

• Numbers are "relatively prime"
• LCM = multiply the numbers together

Using GCF to Simplify Fractions

Example: Simplify 24/36

Step 1: Find GCF of 24 and 36 GCF = 12

Step 2: Divide both by GCF 24 ÷ 12 = 2 36 ÷ 12 = 3

Answer: 24/36 = 2/3 ✓

Using LCM to Add Fractions

Example: Add 1/4 + 1/6

Step 1: Find LCM of 4 and 6 LCM = 12

Step 2: Convert both to denominator of 12 1/4 = 3/12 1/6 = 2/12

Step 3: Add 3/12 + 2/12 = 5/12 ✓

Practice Examples

Example 1: GCF of 18 and 24

Factors of 18: 1, 2, 3, 6, 9, 18 Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24

Common factors: 1, 2, 3, 6

GCF = 6 ✓

Example 2: LCM of 6 and 8

Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48... Multiples of 8: 8, 16, 24, 32, 40, 48...

LCM = 24 ✓

Example 3: GCF and LCM of 10 and 15

GCF: Factors of 10: 1, 2, 5, 10 Factors of 15: 1, 3, 5, 15 GCF = 5 ✓

LCM: Multiples of 10: 10, 20, 30, 40, 50, 60... Multiples of 15: 15, 30, 45, 60... LCM = 30 ✓

The Ladder Method (Alternative)

You can use the "ladder" or "cake" method for both GCF and LCM!

Example: GCF of 24 and 36

Draw a ladder and divide by common factors:

• Divide both by 2 → get 12 and 18
• Divide both by 2 again → get 6 and 9
• Divide both by 3 → get 2 and 3

Multiply the numbers on the left: 2 × 2 × 3 = 12 ✓ (GCF)

For LCM: Multiply left AND bottom: 2 × 2 × 3 × 2 × 3 = 36 ✓

Common Mistakes

❌ Mistake 1: Confusing GCF and LCM GCF is smaller, LCM is bigger!

❌ Mistake 2: Picking a common factor that's not the greatest Make sure you find the BIGGEST common factor!

❌ Mistake 3: Listing multiples incorrectly Double-check you're multiplying correctly (4, 8, 12, 16... not 4, 5, 6, 7...)

Quick Check Method

For GCF: Can you divide both numbers by your answer? ✓ For LCM: Can both numbers divide evenly into your answer? ✓

Tips for Success

Finding GCF: ✅ List factors from smallest to largest ✅ Circle common ones ✅ Pick the biggest!

Finding LCM: ✅ List multiples in order ✅ Find the first one that appears in both lists ✅ That's your LCM!

Summary

GCF (Greatest Common Factor):

• Largest number that divides into both numbers
• Find by: listing factors or using prime factorization
• Smaller than or equal to the numbers
• Use for: simplifying fractions, dividing into equal groups

LCM (Least Common Multiple):

• Smallest number that both numbers divide into
• Find by: listing multiples or using prime factorization
• Larger than or equal to the numbers
• Use for: adding fractions, finding when events repeat

Remember:

• GCF Goes into (dividing)
• LCM is Larger (multiplying)

Both are super useful for working with fractions! ✓