Factors and Multiples

A Number Mystery!

Imagine you have 24 cupcakes to share equally among your friends. How many ways can you divide them so everyone gets the same amount with none left over? The answer involves a mathematical concept that helps us solve puzzles like this every day!

What Are Factors and Multiples?

Factors are numbers that divide evenly into another number with no remainder. Think of them as the “building blocks” that multiply together to make a number. For example, 3 and 4 are factors of 12 because 3 × 4 = 12.

Multiples are the result when you multiply a number by whole numbers (1, 2, 3, 4…). Think of them as the “times table” for that number. The multiples of 5 are 5, 10, 15, 20, 25…

The Key Difference

• Factors divide INTO a number (factors are usually smaller)
• Multiples come FROM multiplying a number (multiples are usually larger)
• Every number has a limited set of factors
• Every number has infinite multiples

Quick check: For the number 12:

• Factors: 1, 2, 3, 4, 6, 12 (only 6 factors)
• Multiples: 12, 24, 36, 48, 60… (infinite!)

Visual Understanding

Factors as arrays: The number 12 can be arranged in different rectangular arrays:

• 1 × 12 (1 row of 12)
• 2 × 6 (2 rows of 6)
• 3 × 4 (3 rows of 4)

These pairs are factor pairs of 12!

Multiples as skip counting: Multiples of 3 are what you get when you skip count by 3: 3, 6, 9, 12, 15, 18, 21…

Teacher’s Insight

Here’s what I’ve learned from teaching: The biggest confusion happens because factors and multiples sound similar but are opposites! I teach students: “Factors go IN (division), Multiples come OUT (multiplication).”

My top tip: Use the “Factor Pairs Method” - always write factors in pairs that multiply to give your number. This way, you never miss a factor!

Classroom breakthrough: When students realize that finding factors is like asking “What multiplication facts make this number?”, everything clicks. For 18, they think: “1×18, 2×9, 3×6” and they’ve found all the factors!

Strategies for Finding Factors and Multiples

Strategy 1: The Factor Pairs Method

Test numbers from 1 upward, finding pairs that multiply to your target number.

Example: Find factors of 20

• 1 × 20 = 20 ✓ (1 and 20 are factors)
• 2 × 10 = 20 ✓ (2 and 10 are factors)
• 3 × ? = 20 ✗ (doesn’t work evenly)
• 4 × 5 = 20 ✓ (4 and 5 are factors)
• 5 × 4 (already found)

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

Strategy 2: The Division Test

Divide your number by each whole number starting from 1. If there’s no remainder, it’s a factor!

Example: Is 7 a factor of 35? 35 ÷ 7 = 5 (no remainder) ✓ Yes!

Strategy 3: Skip Counting for Multiples

To find multiples, multiply your number by 1, 2, 3, 4, 5…

Example: First 5 multiples of 6:

• 6 × 1 = 6
• 6 × 2 = 12
• 6 × 3 = 18
• 6 × 4 = 24
• 6 × 5 = 30

Strategy 4: The “Half-Way” Shortcut for Factors

You only need to check factors up to half the number (or actually up to the square root, but half is easier for kids!).

Example: For 36, you only need to check up to 18, because any factor larger than 18 would pair with something smaller than 2.

Strategy 5: Common Factors Using Lists

To find common factors, list all factors of each number and circle the ones that appear in both lists.

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: 1, 2, 3, 6

Key Vocabulary

• Factor: A number that divides evenly into another number
• Multiple: The result of multiplying a number by a whole number
• Factor Pair: Two factors that multiply together to give the original number
• Common Factor: A factor shared by two or more numbers
• Prime Number: A number with exactly two factors (1 and itself)
• Composite Number: A number with more than two factors
• Divisible: Can be divided evenly with no remainder

Worked Examples

Example 1: Finding All Factors

Problem: List all factors of 18

Solution: 1, 2, 3, 6, 9, 18

Detailed Explanation: Test each number:

• 1 × 18 = 18 ✓ (factors: 1, 18)
• 2 × 9 = 18 ✓ (factors: 2, 9)
• 3 × 6 = 18 ✓ (factors: 3, 6)
• 4 doesn’t divide evenly ✗
• 5 doesn’t divide evenly ✗
• 6 × 3 already found

Think about it: Notice we found factors in pairs? This is the smart way - once you’ve checked up to the middle, you’re done!

Example 2: Listing Multiples

Problem: List the first 5 multiples of 7

Solution: 7, 14, 21, 28, 35

Detailed Explanation: Multiply 7 by 1, 2, 3, 4, 5:

• 7 × 1 = 7
• 7 × 2 = 14
• 7 × 3 = 21
• 7 × 4 = 28
• 7 × 5 = 35

Think about it: Multiples are just the times table! The multiples of 7 are exactly the 7 times table.

Example 3: Factor or Multiple?

Problem: Is 7 a factor or multiple of 35?

Solution: Factor

Detailed Explanation:

• Test: 35 ÷ 7 = 5 (divides evenly)
• Since 7 divides INTO 35, it’s a factor
• (35 is a multiple OF 7, but 7 is a factor OF 35)

Think about it: The smaller number (7) is usually the factor, the larger number (35) is usually the multiple!

Example 4: Finding Common Factors

Problem: Find the common factors of 12 and 18

Solution: 1, 2, 3, 6

Detailed Explanation:

• Factors of 12: 1, 2, 3, 4, 6, 12
• Factors of 18: 1, 2, 3, 6, 9, 18
• Numbers in both lists: 1, 2, 3, 6

Think about it: Common factors are numbers that divide evenly into BOTH numbers. They’re shared factors!

Example 5: Common Multiples

Problem: Find the first 3 common multiples of 4 and 6

Solution: 12, 24, 36

Detailed Explanation:

• Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36…
• Multiples of 6: 6, 12, 18, 24, 30, 36…
• Numbers in both lists: 12, 24, 36

Think about it: Common multiples appear in both times tables!

Example 6: Real-Life Factor Problem

Problem: You have 24 cookies. What are all the ways to arrange them in equal rows?

Solution: 1×24, 2×12, 3×8, 4×6, 6×4, 8×3, 12×2, 24×1

Detailed Explanation: Find all factor pairs of 24:

• 1 and 24 (1 row of 24, or 24 rows of 1)
• 2 and 12 (2 rows of 12, or 12 rows of 2)
• 3 and 8 (3 rows of 8, or 8 rows of 3)
• 4 and 6 (4 rows of 6, or 6 rows of 4)

Think about it: Each factor pair represents a different way to arrange the cookies in a rectangle!

Example 7: Real-Life Multiple Problem

Problem: Bus A arrives every 15 minutes. When will it arrive in the next 2 hours?

Solution: 15, 30, 45, 60, 75, 90, 105, 120 minutes (or 0:15, 0:30, 0:45, 1:00, 1:15, 1:30, 1:45, 2:00)

Detailed Explanation: Find multiples of 15 up to 120: 15, 30, 45, 60, 75, 90, 105, 120

Think about it: The arrival times are multiples of 15! This pattern helps predict when events repeat.

Common Misconceptions & How to Avoid Them

Misconception 1: “Factors and multiples are the same thing”

The Truth: They’re opposites! Factors divide INTO a number, multiples are what you GET when you multiply.

How to think about it correctly: For 12: factors are 1,2,3,4,6,12 (all smaller or equal). Multiples are 12,24,36… (all larger or equal).

Misconception 2: “1 is not a factor”

The Truth: 1 is a factor of every number! (Because 1 × number = number)

How to think about it correctly: Always start your factor list with 1, and end with the number itself.

Misconception 3: “Multiples must be bigger than the original number”

The Truth: The number itself is its first multiple! The first multiple of 5 is 5 (because 5 × 1 = 5).

How to think about it correctly: Multiples start from the number itself and go up infinitely.

Common Errors to Watch Out For

Memory Aids & Tricks

The Factor-Multiple Rhyme

“Factors go IN, small and neat, Multiples grow, they repeat! Factors divide with no remainder left, Multiples multiply - they’re deft!”

The FEW and MANY Rule

Factors are FEW (limited number) Multiples are MANY (infinite)

The “IN vs OUT” Trick

Factors go IN (division) Multiples come OUT (multiplication)

The Factor Pairs Rainbow

When listing factors, make a “rainbow” connecting pairs:

1  2  3  4  6  12
└──┴──┴──┘
   12  6  4

This helps you not miss any!

Practice Problems

Easy Level (Building Understanding)

1. List all factors of 10 Answer: 1, 2, 5, 10 Hint: Think: What multiplies to make 10?

2. List the first 4 multiples of 5 Answer: 5, 10, 15, 20 Hint: Just the 5 times table!

3. Is 3 a factor of 15? Answer: Yes (because 15 ÷ 3 = 5) Hint: Does 3 divide evenly into 15?

4. Is 20 a multiple of 4? Answer: Yes (because 4 × 5 = 20) Hint: Is 20 in the 4 times table?

Medium Level (Apply Your Knowledge)

5. List all factors of 24 Answer: 1, 2, 3, 4, 6, 8, 12, 24 Hint: Work in pairs: 1×24, 2×12, 3×8, 4×6

6. Find the first 3 common multiples of 3 and 4 Answer: 12, 24, 36 Hint: List multiples of each until you find matches

7. What are the common factors of 20 and 30? Answer: 1, 2, 5, 10 Hint: List all factors of each, then find matches

8. Which is greater: the number of factors of 16, or the number of factors of 15? Answer: Factors of 16 (five factors: 1,2,4,8,16 vs four factors: 1,3,5,15) Hint: Count them carefully!

Challenge Level (Think Deeper!)

9. Find a number that has exactly 3 factors Answer: 4, 9, 25, or any perfect square of a prime number Hint: Only perfect squares of primes have exactly 3 factors!

10. You have 36 chairs to arrange in equal rows. List all possible arrangements (factor pairs). Answer: 1×36, 2×18, 3×12, 4×9, 6×6, 9×4, 12×3, 18×2, 36×1 Hint: Find all factor pairs of 36

Real-World Applications

Organizing Events

Scenario: You’re organizing 48 students into equal teams for sports day. What are your options?

Solution: Find factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48

Why this matters: You could make 2 teams of 24, 3 teams of 16, 4 teams of 12, 6 teams of 8, etc. Factors help with fair grouping!

Packaging Products

Scenario: A factory makes 60 chocolate bars per batch. They need to pack them in equal boxes. What box sizes work?

Solution: Factors of 60 are: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60

Why this matters: They could use boxes of 5, 10, 12, or any factor size. Understanding factors prevents waste!

Bus Schedules

Scenario: Two buses leave the station - Bus A every 12 minutes, Bus B every 18 minutes. When do they leave together again?

Solution: Find common multiples of 12 and 18: 36, 72, 108… They’ll both leave together every 36 minutes!

Why this matters: Understanding multiples helps predict when repeating events coincide!

Garden Planning

Scenario: You have 24 plants to arrange in a rectangular garden. What rectangular arrangements are possible?

Solution: Factor pairs of 24: 1×24, 2×12, 3×8, 4×6

Why this matters: You could plant in 4 rows of 6, 3 rows of 8, etc. Factors determine possible layouts!

Music and Rhythm

Scenario: In music, notes repeat in patterns. If a drum beats every 4 counts and a cymbal every 6 counts, when do they sound together?

Solution: Common multiples of 4 and 6: 12, 24, 36… They sound together every 12 counts!

Why this matters: Musicians use multiples to create rhythmic patterns and harmonies!

Study Tips for Mastering Factors and Multiples

1. Practice Times Tables

Knowing multiplication facts makes finding factors and multiples much easier!

2. Use the Factor Rainbow Method

Always work in pairs - it’s faster and you won’t miss any factors.

3. Make Lists

Get comfortable writing out factor lists and multiple lists systematically.

4. Look for Patterns

Notice that multiples increase by the same amount each time (5, 10, 15, 20… goes up by 5).

5. Use Real Objects

Practice with real items - how many ways can you arrange 12 counters in equal groups?

6. Remember: Factors are Finite, Multiples are Infinite

This helps you know when to stop looking for factors!

7. Connect to Division

If you can divide evenly, it’s a factor. Practice your division!

How to Check Your Answers

1. For Factors - Test Each One: If you think 6 is a factor of 24, check: 24 ÷ 6 = 4 ✓ (no remainder)

2. For Factors - Use Multiplication: Does 3 × something = 24? If yes (3×8=24), then 3 is a factor ✓

3. For Multiples - Use Division: Is 35 a multiple of 7? Check: 35 ÷ 7 = 5 (whole number) ✓

4. Count Your Factors: Did you start with 1 and end with the number itself? Both are always factors!

5. Check Factor Pairs: Every factor pairs with another: if you have 3, you should have 8 (because 3×8=24)

Extension Ideas for Fast Learners

Challenge 1: Factor Patterns

Investigate: Do all even numbers have 2 as a factor? Do all numbers ending in 5 have 5 as a factor?

Challenge 2: Perfect Numbers

A perfect number equals the sum of its factors (excluding itself). Example: 6 = 1+2+3. Find the next perfect number after 6!

Challenge 3: Prime Factorization

Learn to break numbers down into prime factors only. Example: 12 = 2 × 2 × 3

Challenge 4: LCM and GCF

Learn about Least Common Multiple (smallest common multiple) and Greatest Common Factor (largest common factor).

Challenge 5: Factor Trees

Create factor trees to find all prime factors of larger numbers like 72 or 100.

Challenge 6: Number Puzzles

“I’m thinking of a number with exactly 4 factors. It’s less than 20. What could it be?” (Answers: 6, 8, 10, 14, 15)

Parent & Teacher Notes

Building Understanding: Use concrete materials! Give students 24 counters and have them find all ways to arrange them in equal groups. This makes factors tangible.

Common Struggles: If students struggle:

• Check their multiplication and division fluency
• Ensure they understand “divides evenly” means “no remainder”
• Practice the difference: “factor OF” vs “multiple OF”
• Use the same numbers repeatedly until patterns emerge

Differentiation Tips:

• Struggling learners: Start with smaller numbers (6, 8, 10, 12). Use physical objects to model factors. Focus on one concept at a time.
• On-track learners: Work with numbers up to 50. Practice both factors and multiples. Include word problems.
• Advanced learners: Explore prime factorization, investigate number properties, work with numbers above 100.

Real-World Connections:

• Packaging and grouping problems
• Sports team formation
• Music rhythm patterns
• Tile and pattern designs
• Schedule coordination

Assessment Ideas:

• “List all factors of 30” - tests systematic thinking
• “Find 3 common multiples of 4 and 5” - tests both concepts
• “You have 18 cookies. How many ways can you share them equally?” - tests application
• “Is 42 a factor or multiple of 7?” - tests understanding of definitions

Common Teaching Pitfalls:

• Moving too quickly between factors and multiples (teach one thoroughly first)
• Not emphasizing the pair method for finding factors
• Forgetting to connect to times tables (multiples ARE times tables!)
• Not using enough real-world contexts

Vocabulary Emphasis: Make sure students can explain the difference in their own words: “A factor divides into a number” vs “A multiple is what you get when you multiply.”

Links to Future Learning: Factors and multiples are foundational for:

• Simplifying fractions (use GCF)
• Finding common denominators (use LCM)
• Prime factorization
• Algebraic factoring
• Divisibility rules

Remember: Factors and multiples are opposite sides of the same coin! Master this relationship, and many other math concepts become easier.