Factors and Multiples

For Elementary Students

What Are Factors?

Factors are numbers that divide evenly into another number (no remainder left over).

Think about it like this: Factors are the numbers you multiply together to GET a number!

Example: Factors of 12

What numbers can you multiply to make 12?

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

So the factors of 12 are: 1, 2, 3, 4, 6, 12

Finding Factors

Example: Find all factors of 20

Try dividing 20 by each number:

• 20 ÷ 1 = 20 ✓ (1 is a factor)
• 20 ÷ 2 = 10 ✓ (2 is a factor)
• 20 ÷ 3 = 6.67... ❌ (not a factor — has a remainder)
• 20 ÷ 4 = 5 ✓ (4 is a factor)
• 20 ÷ 5 = 4 ✓ (5 is a factor)

Factors of 20: 1, 2, 4, 5, 10, 20

What Are Multiples?

Multiples are what you get when you count by a number!

Example: Multiples of 3

Count by 3s: 3, 6, 9, 12, 15, 18, 21, 24, ...

These are all multiples of 3!

Example: Multiples of 5

Count by 5s: 5, 10, 15, 20, 25, 30, ...

Factors vs. Multiples — What's the Difference?

Factors go INTO a number (smaller or equal):

• 3 is a factor of 12 (3 goes into 12)

Multiples come FROM a number (larger or equal):

• 12 is a multiple of 3 (3 × 4 = 12)

Memory trick: Think "Factors are Fewer, Multiples are Many!"

Important Fact

• A number has a LIMITED number of factors (you can list them all)
• A number has INFINITE multiples (they go on forever!)

For Junior High Students

Understanding Factors

Factors of a number are all whole numbers that divide into it evenly (remainder = 0).

Example: Factors of 24

Pairs that multiply to 24:

• 1 × 24
• 2 × 12
• 3 × 8
• 4 × 6

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

Pattern: Factors come in pairs! When you find one, you find two.

Understanding Multiples

Multiples are the results of multiplying a number by whole numbers (1, 2, 3, 4, ...).

Example: Multiples of 7

• 7 × 1 = 7
• 7 × 2 = 14
• 7 × 3 = 21
• 7 × 4 = 28

Multiples of 7: 7, 14, 21, 28, 35, ...

Key Differences

Greatest Common Divisor (GCD)

GCD (also called GCF — Greatest Common Factor) is the largest number that divides both numbers evenly.

Example: Find GCD of 18 and 24

Step 1: List all factors

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

Step 2: Find common factors: 1, 2, 3, 6

Step 3: Pick the largest: GCD = 6

Using GCD

Simplifying fractions: Divide both numerator and denominator by the GCD

Example: Simplify 18/24

• GCD(18, 24) = 6
• 18 ÷ 6 = 3
• 24 ÷ 6 = 4
• Simplified: 3/4

Least Common Multiple (LCM)

LCM is the smallest number that is a multiple of both numbers.

Example: Find LCM of 4 and 6

Step 1: List multiples

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

Step 2: Find common multiples: 12, 24, 36, ...

Step 3: Pick the smallest: LCM = 12

Using LCM

Adding fractions: Find a common denominator

Example: Add 1/4 + 1/6

• LCM(4, 6) = 12
• Convert: 1/4 = 3/12 and 1/6 = 2/12
• Add: 3/12 + 2/12 = 5/12

Finding GCD Using Prime Factorization

Example: GCD of 18 and 24

Prime factorizations:

• 18 = 2 × 3 × 3
• 24 = 2 × 2 × 2 × 3

Common factors: 2 and 3

GCD = 2 × 3 = 6

Finding LCM Using Prime Factorization

Example: LCM of 18 and 24

Prime factorizations:

• 18 = 2 × 3 × 3
• 24 = 2 × 2 × 2 × 3

Take highest power of each prime:

• 2³ (from 24) = 8
• 3² (from 18) = 9

LCM = 2³ × 3² = 8 × 9 = 72

Special Cases

Coprime numbers: GCD = 1 (share no common factors except 1)

• Example: GCD(7, 10) = 1

When one divides the other:

• GCD equals the smaller number
• LCM equals the larger number
• Example: GCD(6, 18) = 6, LCM(6, 18) = 18

Relationship Between GCD and LCM

For any two numbers a and b:

GCD(a, b) × LCM(a, b) = a × b

Example: a = 12, b = 18

• GCD(12, 18) = 6
• LCM(12, 18) = 36
• Check: 6 × 36 = 216 and 12 × 18 = 216 ✓

Real-Life Applications

GCD:

• Simplifying fractions
• Dividing things into equal groups
• Reducing ratios

LCM:

• Finding common denominators
• Scheduling (events that repeat)
• Buying items in packages

Example: Two buses leave at the same time. One returns every 12 minutes, the other every 18 minutes. When do they meet again?

LCM(12, 18) = 36 minutes

Practice