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Teacher Guide: Factors and Multiples

Learn what factors and multiples are, how to find them, and why they're important in math.

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10 questions on Factors & Multiples. Students join with a name, you see everyone's score.

For Teachers

Learning Objectives
  • Define factors as numbers that divide evenly into another number
  • Define multiples as products of a number and whole numbers
  • Find all factors of a given number
  • List multiples of a given number
  • Distinguish between factors and multiples
  • Apply factors and multiples to real-world situations
Prerequisites
  • • Multiplication facts through 12
  • • Basic division with whole numbers
  • • Understanding of remainders
Discussion Starters
  • 1. Why does every number greater than 1 have at least two factors (1 and itself)? What about the number 1?
  • 2. Can a small number be a multiple of a larger number? Why or why not?
  • 3. How are factors and multiples related to multiplication and division?
  • 4. When might you need to find factors or multiples in real life?
Common Misconceptions

Factors are always smaller than the number

Remediation: The number itself is always a factor! 12 is a factor of 12 because 12÷12=112 \div 12 = 1.

Multiples must be larger than the number

Remediation: The first multiple of any number is the number itself: the first multiple of 5 is 5.

Factor pairs are always different numbers

Remediation: Perfect squares have a repeated factor: the factor pairs of 36 include (6,6)(6, 6).

Differentiation Ideas

For Struggling Students:

  • • Use multiplication tables to find factors
  • • Start with small numbers (under 20)
  • • Use physical manipulatives to show grouping
  • • Create a factors vs multiples anchor chart

For On-Level Students:

  • • Find all factors of two-digit numbers
  • • Identify common factors of two numbers
  • • Solve word problems involving equal groups

For Advanced Students:

  • • Explore the relationship to prime factorization
  • • Investigate perfect numbers (the factors other than the number itself add up to the number: 6=1+2+36 = 1 + 2 + 3)
  • • Find GCF and LCM using factors and multiples
Standards Alignment
  • 4.OA.B.4 (CCSS.MATH.CONTENT.4.OA.B.4)

    Find all factor pairs for a whole number in the range 1-100. Recognize that a whole number is a multiple of each of its factors.

  • 6.NS.B.4 (CCSS.MATH.CONTENT.6.NS.B.4)

    Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12.

Lesson Resources
  • visualFactor Tree Builder

    Students build factor trees to break a number into factors

  • activityFactor vs Multiple Sorting Game

    Sort numbers into factor or multiple categories

  • worksheetReal-World Factors

    Solve problems about sharing, arranging, and scheduling

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

Definition

Factors are numbers that divide evenly into another number with no remainder.
Multiples are the results of multiplying a number by whole numbers.
Example with the number 12:
Factors of 12: 1,2,3,4,6,121, 2, 3, 4, 6, 12
  • 12÷1=1212 \div 1 = 12 (no remainder)
  • 12÷2=612 \div 2 = 6 (no remainder)
  • 12÷3=412 \div 3 = 4 (no remainder)
  • 12÷4=312 \div 4 = 3 (no remainder)
  • 12÷6=212 \div 6 = 2 (no remainder)
  • 12÷12=112 \div 12 = 1 (no remainder)
Multiples of 12: 12,24,36,48,60,…12, 24, 36, 48, 60, \ldots
  • 12×1=1212 \times 1 = 12
  • 12×2=2412 \times 2 = 24
  • 12×3=3612 \times 3 = 36
  • 12×4=4812 \times 4 = 48
  • 12×5=6012 \times 5 = 60

Worked Examples

Find all the factors of 18.
1

Start with 1

18÷1=1818 \div 1 = 18, so 1 and 18 are factors → Factors: 1,181, 18
2

Try 2

18÷2=918 \div 2 = 9, so 2 and 9 are factors → Factors: 1,2,9,181, 2, 9, 18
3

Try 3

18÷3=618 \div 3 = 6, so 3 and 6 are factors → Factors: 1,2,3,6,9,181, 2, 3, 6, 9, 18
4

Try 4

18÷4=4.518 \div 4 = 4.5 (not a whole number) → 4 is NOT a factor
5

Stop at the square root

18≈4.24\sqrt{18} \approx 4.24, and the next number to try, 5, is already bigger, so every factor has been found → All factors: 1,2,3,6,9,181, 2, 3, 6, 9, 18

Common Mistakes

Confusing factors and multiples

Why it's wrong: The words sound similar but mean opposite things. Factors divide INTO a number; multiples are made BY multiplying.

Correct: Remember: Factors are Few (limited), Multiples are Many (infinite). 12 has 6 factors, but infinite multiples.

Forgetting that 1 is always a factor

Why it's wrong: Every number divided by 1 gives itself, so 1 is always a factor.

Correct: Always start your factor list with 1, and don't forget the number itself is also a factor.

Starting a list of multiples with 0

Why it's wrong: n×0=0n \times 0 = 0, so 0 is a multiple of every number. But when we list multiples in class, we start at n×1n \times 1.

Correct: Start the list with the number itself: the multiples of 5 are 5,10,15,…5, 10, 15, \ldots

Missing factor pairs

Why it's wrong: When finding factors, students sometimes forget that factors come in pairs.

Correct: When you find one factor, find its pair: if 18÷2=918 \div 2 = 9, then both 2 AND 9 are factors.

Why It Matters

Factors and multiples help you in many ways:
  • Simplifying fractions: Find common factors to reduce 812\frac{8}{12} to 23\frac{2}{3}
  • Adding fractions: Find common multiples to get the same denominator
  • Sharing equally: Divide 24 cookies among 6 friends (24 is divisible by 6)
  • Patterns: Understand why every even number has 2 as a factor
  • Problem solving: Figure out schedules, arrangements, and groupings
These skills are essential for algebra, fractions, and real-world math!

Real World Applications

Party Planning

When organizing events, you need to divide things equally among guests.

Example:

You have 30 cupcakes. Which group sizes can share them equally? The factors of 30: 1,2,3,5,6,10,15,301, 2, 3, 5, 6, 10, 15, 30. So you can have groups of 2, 3, 5, 6, 10, 15, or 30 people.

1Try It Yourself

You're planning a party with 24 party favors to distribute equally.

How many guests can you invite so everyone gets the same number of favors?

Step 1: Write the mathematical expression

Find the factors of 24:

Scheduling and Patterns

Multiples help us find when events occur together.

Example:

Bus A comes every 15 minutes. Bus B comes every 20 minutes. When do they arrive together? Find common multiples of 15 and 20: 60,120,180,…60, 120, 180, \ldots They arrive together every 60 minutes.

2Try It Yourself

You practice piano every 3 days and soccer every 4 days.

If you have both today, in how many days will you have both again?

Step 1: Write the mathematical expression

Find the first common multiple of 3 and 4:

Arranging Objects

Factors help determine how to arrange items in rows and columns.

Example:

You have 48 chairs to arrange. Possible arrangements: 1 row of 48, 2 rows of 24, 3 rows of 16, 4 rows of 12, 6 rows of 8, 8 rows of 6, etc.

3Try It Yourself

A classroom has 28 desks to arrange in equal rows.

What are all the possible arrangements?

Step 1: Write the mathematical expression

Find the factor pairs of 28, written like (1, 28):

Key Takeaways

  • 1Factors divide evenly into a number with no remainder
  • 2Multiples are made by multiplying a number by 1, 2, 3, ...
  • 3Every number has 1 and itself as factors
  • 4Factors come in pairs that multiply to give the original number
  • 5Multiples are infinite, but factors are finite
  • 6Factors help with division and simplifying; multiples help with patterns and scheduling

Frequently Asked Questions

Is a number a factor and multiple of itself?

Yes! Every number is both a factor of itself (12÷12=112 \div 12 = 1) and a multiple of itself (12×1=1212 \times 1 = 12).

How do I know when I've found all the factors?

Stop checking when your divisor passes the square root. For 36, 36=6\sqrt{36} = 6, so after checking up to 6, you've found all factor pairs.

What's the difference between factors and divisors?

They mean the same thing! 'Factor' and 'divisor' are two words for numbers that divide evenly into another number.

Glossary

Factor
A number that divides evenly into another number with no remainder
Multiple
The product of a number and any whole number
Factor pair
Two numbers that multiply together to give a specific product
Divisible
A number is divisible by another if the division has no remainder
Prime number
A number with exactly two factors: 1 and itself

More in This Topic

Prime Factorization

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Prime and Composite Numbers

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Greatest Common Factor (GCF)

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Least Common Multiple (LCM)

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