How to Explain Factors and Multiples to Fourth Graders

"Is 7 a factor of 42?"

"Is 42 a multiple of 7?"

These questions ask about the same relationship from different directions. Fourth grade introduces this powerful lens for understanding numbers—factors and multiples reveal structure that helps with fractions, division, and algebra.

Why Factors and Multiples Matter

Understanding factors and multiples helps students:

• Find common denominators for fractions
• Simplify fractions to lowest terms
• Understand divisibility rules
• Solve problems about grouping and sharing
• Prepare for greatest common factor and least common multiple

These concepts form the foundation of number theory and algebraic thinking.

What Are Factors?

The Definition

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

For 12:

• 12 ÷ 1 = 12 ✓ (no remainder, so 1 is a factor)
• 12 ÷ 2 = 6 ✓ (no remainder, so 2 is a factor)
• 12 ÷ 3 = 4 ✓ (no remainder, so 3 is a factor)
• 12 ÷ 4 = 3 ✓ (no remainder, so 4 is a factor)
• 12 ÷ 5 = 2.4 ✗ (has remainder, so 5 is NOT a factor)
• 12 ÷ 6 = 2 ✓ (no remainder, so 6 is a factor)
• 12 ÷ 12 = 1 ✓ (no remainder, so 12 is a factor)

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

Factors Come in Pairs

Factors pair up to make the number:

Factors of 12:
1 × 12 = 12
2 × 6 = 12
3 × 4 = 12

Factor pairs: (1,12), (2,6), (3,4)

This pairing helps you find all factors systematically—once you find one, divide to find its partner.

Finding All Factors

Method: Start at 1 and work up

Find all factors of 36:

1. 36 ÷ 1 = 36 → Factors: 1, 36
2. 36 ÷ 2 = 18 → Factors: 2, 18
3. 36 ÷ 3 = 12 → Factors: 3, 12
4. 36 ÷ 4 = 9 → Factors: 4, 9
5. 36 ÷ 5 = 7.2 → Not a factor
6. 36 ÷ 6 = 6 → Factors: 6, 6 (same number—we're done!)

Stop when factor pairs start repeating or when you reach the same number twice.

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

Important Facts About Factors

1. Every number has at least two factors: 1 and itself
2. Factors are FINITE: There's a limited list
3. The largest factor is the number itself
4. The smallest factor is always 1

What Are Multiples?

The Definition

A multiple is the result of multiplying a number by any whole number.

Multiples of 3:

• 3 × 1 = 3
• 3 × 2 = 6
• 3 × 3 = 9
• 3 × 4 = 12
• 3 × 5 = 15
• ...and so on forever!

Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, ...

Multiples Are Infinite

Unlike factors, multiples go on forever. You can always multiply by the next whole number to get another multiple.

Skip Counting IS Finding Multiples

When students skip count by 5: 5, 10, 15, 20, 25, 30...

They're listing multiples of 5!

The Multiplication Table Connection

Every number in the times table row for 4 is a multiple of 4:

4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40...

Factors vs. Multiples: The Key Distinction

Factors go INTO a number:

• "3 is a factor of 12" means 3 divides into 12 evenly

Multiples come FROM a number:

• "12 is a multiple of 3" means 12 = 3 × something

The Same Relationship, Different Directions

If 3 is a factor of 12, then 12 is a multiple of 3.

These are two ways to describe the same relationship:

• 3 × 4 = 12
• Factor view: 3 and 4 are both factors of 12
• Multiple view: 12 is a multiple of both 3 and 4

Memory Trick

• Factors: Think "fits into" (factors FIT INTO numbers)
• Multiples: Think "multiplied" (multiples come from multiplying)

Prime and Composite Numbers

Prime Numbers

A prime number has exactly TWO factors: 1 and itself.

2: factors are 1, 2 → Prime!
3: factors are 1, 3 → Prime!
5: factors are 1, 5 → Prime!
7: factors are 1, 7 → Prime!

The first several prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29...

Note: 2 is the only even prime number!

Composite Numbers

A composite number has MORE than two factors.

4: factors are 1, 2, 4 → Composite (3 factors)
6: factors are 1, 2, 3, 6 → Composite (4 factors)
9: factors are 1, 3, 9 → Composite (3 factors)

What About 1?

1 is neither prime nor composite.

1 has only ONE factor (itself), so it doesn't fit either definition:

• Not prime (needs exactly 2 factors)
• Not composite (needs more than 2 factors)

How to Test if a Number is Prime

Check if any number from 2 up to the square root of the number divides it evenly.

Is 17 prime?

• 17 ÷ 2 = 8.5 (not even)
• 17 ÷ 3 = 5.67 (not divisible)
• 17 ÷ 4 = 4.25 (not divisible)
• Since √17 ≈ 4.1, we can stop

17 has no factors besides 1 and 17, so it's prime!

Is 15 prime?

• 15 ÷ 3 = 5 ✓

15 has a factor besides 1 and 15, so it's composite.

Divisibility Rules (Shortcuts!)

Divisible by 2

Last digit is even (0, 2, 4, 6, 8)

• 456 → 6 is even → divisible by 2 ✓

Divisible by 3

Sum of digits is divisible by 3

• 456 → 4 + 5 + 6 = 15 → 15 ÷ 3 = 5 → divisible by 3 ✓

Divisible by 4

Last two digits form a number divisible by 4

• 456 → 56 ÷ 4 = 14 → divisible by 4 ✓

Divisible by 5

Last digit is 0 or 5

• 456 → ends in 6 → NOT divisible by 5

Divisible by 6

Divisible by BOTH 2 and 3

• 456 → divisible by 2 ✓ and by 3 ✓ → divisible by 6 ✓

Divisible by 9

Sum of digits is divisible by 9

• 456 → 4 + 5 + 6 = 15 → 15 ÷ 9 = 1.67 → NOT divisible by 9

Divisible by 10

Last digit is 0

• 456 → ends in 6 → NOT divisible by 10

Hands-On Activities

Factor Rainbows

Draw factors as arcs connecting factor pairs:

        ___________
       /     5     \
      /  ___4___    \
     / /   3   \    \
    | |    |    |    |
    1 2    3    4   6   12

Factor pairs of 12: 1-12, 2-6, 3-4

The rainbow shows how factors pair up!

Array Building

Build arrays to show factors:

12 can be arranged as:

1 × 12: ● ● ● ● ● ● ● ● ● ● ● ●

2 × 6:  ● ● ● ● ● ●
        ● ● ● ● ● ●

3 × 4:  ● ● ● ●
        ● ● ● ●
        ● ● ● ●

4 × 3:  ● ● ●
        ● ● ●
        ● ● ●
        ● ● ●

Each array shows a factor pair!

The Sieve of Eratosthenes

Find all prime numbers 1-100:

1. List numbers 1-100
2. Cross out 1 (neither prime nor composite)
3. Circle 2, cross out all multiples of 2
4. Circle 3, cross out all multiples of 3
5. Circle 5, cross out all multiples of 5
6. Circle 7, cross out all multiples of 7
7. All remaining numbers are prime!

Factor/Multiple Sorting Game

Make cards with numbers. Sort them:

• "Is 24 a multiple of 6?" (Put in YES pile)
• "Is 5 a factor of 24?" (Put in NO pile)

The Venn Diagram Challenge

Create overlapping circles for:

• Multiples of 3
• Multiples of 4

Where do they overlap? (Multiples of 12!)

Common Mistakes and How to Fix Them

Mistake 1: Confusing Factors and Multiples

Wrong: "The multiples of 12 are 1, 2, 3, 4, 6, 12"

Fix: Use the keyword check:

• Factors "go into" (divide evenly into)
• Multiples "come from" (multiply to get)

Multiples of 12 are 12, 24, 36, 48... (skip counting by 12)

Mistake 2: Missing Factor Pairs

Wrong: Listing factors of 24 as "1, 2, 3, 4, 24" (missing 6, 8, 12)

Fix: Use systematic factor pair finding:

• 1 × 24
• 2 × 12
• 3 × 8
• 4 × 6

Mistake 3: Thinking 1 is Prime

Wrong: "1 is prime because its only factors are 1 and itself"

Fix: Clarify: Prime means exactly TWO factors. 1 has only ONE factor (itself counts only once), so it's neither prime nor composite.

Mistake 4: Stopping Too Early When Finding Factors

Wrong: Finding factors of 36 but stopping at 6 × 6, missing that 9 is also a factor

Fix: Check every number up to the point where pairs repeat. 36 ÷ 9 = 4, so 9 is a factor too!

Building Number Sense

Factor Trees

Break composite numbers into prime factors:

       24
      /  \
     4    6
    / \  / \
   2  2 2   3

24 = 2 × 2 × 2 × 3

Looking for Patterns

• What do all multiples of 2 have in common? (Even)
• What do all multiples of 5 have in common? (End in 0 or 5)
• What do all multiples of 10 have in common? (End in 0)

Number Relationships

If 4 is a factor of 20, and 2 is a factor of 4, then 2 must be a factor of 20!

These logical connections build mathematical reasoning.

Connecting to Future Concepts

Factors and multiples prepare students for:

Fractions

• Finding common denominators (multiples)
• Simplifying fractions (common factors)

Greatest Common Factor (GCF)

The largest factor two numbers share

Least Common Multiple (LCM)

The smallest multiple two numbers share

Algebra

Factoring expressions uses the same concepts

Practice Ideas for Home

Factor Hunt

Pick a number. Race to find all its factors. Check by making sure each factor divides evenly.

Skip Counting Games

Skip count by different numbers. What patterns do you notice in the multiples?

Prime Number Challenge

Is this number prime? Test it using divisibility rules and division.

Real-World Factors

"We have 24 cookies to share equally. What are all the ways we can share them?" (Each way uses a factor pair!)

Multiplication Table Exploration

Circle all multiples of 6 on a multiplication table. What patterns do you see?

The Bottom Line

Factors and multiples give students a powerful way to understand numbers—not just as quantities, but as having internal structure.

A number like 24 isn't just "twenty-four." It's:

• 1 × 24
• 2 × 12
• 3 × 8
• 4 × 6
• A multiple of 1, 2, 3, 4, 6, 8, 12, and 24
• A composite number
• 2³ × 3 in prime factorization

When your fourth grader sees numbers this way, they have tools that will serve them through fractions, algebra, and beyond. Factors and multiples are the hidden structure of arithmetic—and discovering them is like learning to see beneath the surface of numbers.