GCF and LCM

For Elementary Students

What is a Factor?

A factor is a number that divides evenly into another number (no remainder).

Example: Factors of 12

• 12 ÷ 1 = 12 ✓
• 12 ÷ 2 = 6 ✓
• 12 ÷ 3 = 4 ✓
• 12 ÷ 4 = 3 ✓
• 12 ÷ 6 = 2 ✓
• 12 ÷ 12 = 1 ✓

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

What is the GCF?

GCF stands for Greatest Common Factor — the largest number that divides evenly into both numbers.

Think about it like this: What's the biggest number that both numbers share as a factor?

Example: Find 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 (factors they both have): 1, 2, 3, 6

GCF = 6 (the greatest one!)

Finding GCF (List Method)

Step 1: List all factors of each number

Step 2: Circle the factors that appear in both lists

Step 3: Pick the largest one

Example: GCF of 8 and 12

Factors of 8: 1, 2, 4, 8 Factors of 12: 1, 2, 3, 4, 6, 12

Common: 1, 2, 4

GCF = 4

What is a Multiple?

A multiple is what you get when you multiply a number by 1, 2, 3, 4, etc.

Example: Multiples of 4

• 4 × 1 = 4
• 4 × 2 = 8
• 4 × 3 = 12
• 4 × 4 = 16

First few multiples of 4: 4, 8, 12, 16, 20, 24...

What is the LCM?

LCM stands for Least Common Multiple — the smallest number that both numbers divide into evenly.

Think about it like this: What's the smallest number that's a multiple of both?

Example: Find LCM of 4 and 6

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

Common multiples: 12, 24, 36...

LCM = 12 (the least/smallest one!)

Finding LCM (List Method)

Step 1: List multiples of each number

Step 2: Find the first (smallest) number in both lists

Step 3: That's the LCM!

Example: LCM of 3 and 5

Multiples of 3: 3, 6, 9, 12, 15, 18... Multiples of 5: 5, 10, 15, 20...

LCM = 15

For Junior High Students

GCF Method: Prime Factorization

Step 1: Write prime factorization of each number

Step 2: Identify common prime factors

Step 3: Multiply the common factors

Example: GCF of 24 and 36

24 = 2 × 2 × 2 × 3 = 2³ × 3 36 = 2 × 2 × 3 × 3 = 2² × 3²

Common factors: 2² (both have at least two 2s) and 3 (both have at least one 3)

GCF = 2² × 3 = 4 × 3 = 12

LCM Method: Prime Factorization

Step 1: Write prime factorization

Step 2: Take the highest power of each prime that appears

Step 3: Multiply them together

Example: LCM of 12 and 18

12 = 2² × 3 18 = 2 × 3²

Take highest powers:

• Highest power of 2: 2² (from 12)
• Highest power of 3: 3² (from 18)

LCM = 2² × 3² = 4 × 9 = 36

GCF and LCM Relationship

For any two numbers a and b:

GCF(a,b) × LCM(a,b) = a × b

Example: GCF and LCM of 6 and 8

• GCF = 2
• LCM = 24
• Check: 2 × 24 = 48 and 6 × 8 = 48 ✓

When GCF = 1

If two numbers have no common factors except 1, they're called relatively prime (or coprime).

Example: 8 and 15

Factors of 8: 1, 2, 4, 8 Factors of 15: 1, 3, 5, 15

GCF = 1 (only common factor)

8 and 15 are relatively prime.

Using GCF to Simplify Fractions

Example: Simplify 24/36

Step 1: Find GCF of 24 and 36 → 12

Step 2: Divide both by GCF

• 24 ÷ 12 = 2
• 36 ÷ 12 = 3

Answer: 24/36 = 2/3 (simplest form)

Using LCM for Adding Fractions

To add fractions with different denominators, you need a common denominator — that's the LCM!

Example: Add 1/4 + 1/6

Step 1: Find LCM of 4 and 6 → 12

Step 2: Convert fractions

• 1/4 = 3/12
• 1/6 = 2/12

Step 3: Add 3/12 + 2/12 = 5/12

Real-World Uses

GCF:

• Arranging items in equal groups
• Cutting boards/ropes into largest equal pieces
• Simplifying fractions

LCM:

• Finding when events repeat together (bus schedules)
• Common denominators for fractions
• Tile patterns that repeat

Example: Two bells ring every 6 minutes and 8 minutes. When do they ring together?

LCM of 6 and 8 = 24 minutes

They ring together every 24 minutes!

Shortcut: When One Number Divides the Other

If one number is a multiple of the other:

GCF = the smaller number LCM = the larger number

Example: 5 and 15

• 15 = 5 × 3 (15 is a multiple of 5)
• GCF = 5
• LCM = 15

Practice