Factors and Multiples

What are Factors?

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

Example: Factors of 12

What numbers divide evenly into 12?

• 12 ÷ 1 = 12 ✓
• 12 ÷ 2 = 6 ✓
• 12 ÷ 3 = 4 ✓
• 12 ÷ 4 = 3 ✓
• 12 ÷ 5 = 2.4 ✗ (not a whole number!)
• 12 ÷ 6 = 2 ✓
• 12 ÷ 12 = 1 ✓

Factors of 12: 1, 2, 3, 4, 6, 12 ✓

Finding Factors - The Pair Method

Find factors in pairs that multiply to make the number!

Example: Factors of 24

1 × 24 = 24 → Factors: 1, 24 2 × 12 = 24 → Factors: 2, 12 3 × 8 = 24 → Factors: 3, 8 4 × 6 = 24 → Factors: 4, 6

All factors of 24: 1, 2, 3, 4, 6, 8, 12, 24 ✓

Tip: Write them in order from smallest to largest!

Factor Rules

Rule 1: Every number has at least TWO factors: 1 and itself

• Factors of 7: 1, 7

Rule 2: 1 is a factor of every number

• 1 is a factor of 5, 10, 99, 1000...

Rule 3: The number itself is always a factor

• 15 is a factor of 15

What are Multiples?

Multiples are what you get when you multiply a number by whole numbers (1, 2, 3, 4...).

Example: Multiples of 3

3 × 1 = 3 3 × 2 = 6 3 × 3 = 9 3 × 4 = 12 3 × 5 = 15...

Multiples of 3: 3, 6, 9, 12, 15, 18, 21... ✓

Important: Multiples go on forever! ∞

Finding Multiples

Just keep multiplying by 1, 2, 3, 4, 5...

Example: First 6 multiples of 5

5 × 1 = 5 5 × 2 = 10 5 × 3 = 15 5 × 4 = 20 5 × 5 = 25 5 × 6 = 30

Answer: 5, 10, 15, 20, 25, 30 ✓

Multiple Rules

Rule 1: Every number is a multiple of itself

• 7 is a multiple of 7 (7 × 1 = 7)

Rule 2: Every number is a multiple of 1

• 1 × 20 = 20, so 20 is a multiple of 1

Rule 3: Multiples get bigger and bigger

• They go to infinity!

Factors vs. Multiples

Think of it this way:

Factors are SMALL (smaller than or equal to the number)

• Factors of 12: 1, 2, 3, 4, 6, 12 (all ≤ 12)

Multiples are BIG (bigger than or equal to the number)

• Multiples of 12: 12, 24, 36, 48... (all ≥ 12)

Factors divide INTO a number Multiples are what you GET when multiplying

Common Factors

Common factors are factors that two numbers share.

Example: Common factors 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 ✓ (appear in both lists)

Greatest Common Factor (GCF)

The largest factor that two numbers share.

Example: GCF of 12 and 18

Common factors: 1, 2, 3, 6

GCF = 6 ✓ (the biggest!)

We'll learn more about GCF in the next lesson!

Common Multiples

Common multiples are multiples that two numbers share.

Example: Common multiples of 3 and 4

Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30... Multiples of 4: 4, 8, 12, 16, 20, 24, 28...

Common multiples: 12, 24, 36... ✓

Least Common Multiple (LCM)

The smallest multiple that two numbers share (not counting 0).

Example: LCM of 3 and 4

LCM = 12 ✓ (the smallest common multiple)

We'll learn more about LCM in the next lesson!

Finding All Factors - Organized Method

Example: Find all factors of 36

Step 1: Start with 1 1 × 36 = 36 → Write: 1, ___, ___, 36

Step 2: Try 2 2 × 18 = 36 → Write: 1, 2, ___, 18, 36

Step 3: Try 3 3 × 12 = 36 → Write: 1, 2, 3, ___, 12, 18, 36

Step 4: Try 4 4 × 9 = 36 → Write: 1, 2, 3, 4, 9, 12, 18, 36

Step 5: Try 5 36 ÷ 5 = 7.2 ✗ Not a factor!

Step 6: Try 6 6 × 6 = 36 → Write: 1, 2, 3, 4, 6, 9, 12, 18, 36

Stop when pairs repeat!

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

Factor Pairs

Factors often come in pairs that multiply together.

Example: Factor pairs of 20

1 × 20 = 20 2 × 10 = 20 4 × 5 = 20

Factor pairs: (1,20), (2,10), (4,5) ✓

Divisibility Tricks

Quick ways to check if a number is a factor:

Factor 2: Ends in 0, 2, 4, 6, or 8 (even)

• 24 → Yes! (ends in 4) ✓

Factor 3: Digit sum divisible by 3

• 24 → 2+4=6, and 6÷3=2 ✓ Yes!

Factor 5: Ends in 0 or 5

• 24 → No (ends in 4) ✗

Factor 10: Ends in 0

• 24 → No ✗

Real-World Examples

Factors in Real Life

Problem: You have 24 cupcakes. What are all the ways you can arrange them in equal rows?

Solution: Find factors of 24!

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

You could make:

• 1 row of 24 cupcakes
• 2 rows of 12 cupcakes
• 3 rows of 8 cupcakes
• 4 rows of 6 cupcakes
• 6 rows of 4 cupcakes
• 8 rows of 3 cupcakes
• 12 rows of 2 cupcakes
• 24 rows of 1 cupcake

Multiples in Real Life

Problem: Hot dogs come in packs of 8. How many hot dogs will you have if you buy 1, 2, 3, or 4 packs?

Solution: Find multiples of 8!

8 × 1 = 8 hot dogs 8 × 2 = 16 hot dogs 8 × 3 = 24 hot dogs 8 × 4 = 32 hot dogs

Practice Strategy

To find factors:

1. Start with 1
2. Test each number: Does it divide evenly?
3. Write factor pairs
4. Stop when pairs repeat
5. List in order

To find multiples:

1. Multiply by 1, 2, 3, 4, 5...
2. Keep going as many as you need
3. Remember they go on forever!

Common Mistakes

❌ Mistake 1: Forgetting 1 and the number itself Wrong: Factors of 10 are 2, 5 ✗ Right: Factors of 10 are 1, 2, 5, 10 ✓

❌ Mistake 2: Mixing up factors and multiples Factors are small, multiples are big!

❌ Mistake 3: Listing factors out of order Always write them from smallest to largest!

Quick Check

Is 4 a factor of 20? 20 ÷ 4 = 5 ✓ Yes! (no remainder)

Is 20 a multiple of 4? 4 × 5 = 20 ✓ Yes!

Both are true! 4 is a factor of 20, AND 20 is a multiple of 4.

Summary

Factors:

• Numbers that divide evenly into another number
• Small (≤ the number)
• Limited amount
• Example: Factors of 12 are 1, 2, 3, 4, 6, 12

Multiples:

• Products of a number multiplied by 1, 2, 3...
• Big (≥ the number)
• Infinite (go on forever)
• Example: Multiples of 12 are 12, 24, 36, 48...

Remember: Factors divide IN, multiples multiply OUT! ✓