Identifying Multiples Worksheet | 4th Grade Math Exercises

Master multiples, patterns, and divisibility: learn to identify multiples, understand skip counting, and solve problems with step-by-step solutions.

Solution: Exercises 1 to 3
1 Finding Multiples
Exercise 1
List the first 8 multiples of 6.
Definition:

Multiple: A number that results from multiplying a given number by any whole number.

Skip Counting: Counting by a specific number (e.g., counting by 6s: 6, 12, 18, 24...).

Method to find multiples:
  1. Multiply the base number by 1, 2, 3, 4, 5, etc.
  2. Or use skip counting (add the base number repeatedly)
  3. List the results in order
Base Number
6
Multipliers
1 to 8
Step 1: Multiply 6 by 1

6 × 1 = 6

Step 2: Multiply 6 by 2

6 × 2 = 12

Step 3: Multiply 6 by 3

6 × 3 = 18

Step 4: Multiply 6 by 4

6 × 4 = 24

Step 5: Multiply 6 by 5

6 × 5 = 30

Step 6: Multiply 6 by 6

6 × 6 = 36

Step 7: Multiply 6 by 7

6 × 7 = 42

Step 8: Multiply 6 by 8

6 × 8 = 48

6
12
18
24
30
36
42
48
6 × 1 = 6
→
6 × 2 = 12
→
6 × 3 = 18
→
6 × 4 = 24
→
6 × 5 = 30
→
6 × 6 = 36
→
6 × 7 = 42
→
6 × 8 = 48
First 8 multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48
Final answer:

The first 8 multiples of 6 are: 6, 12, 18, 24, 30, 36, 42, 48

Applied rules:

• Multiplication Rule: Multiples are found by multiplying the base number by whole numbers

• Pattern Rule: Each multiple is the previous multiple plus the base number

• Order Rule: Multiples are listed in ascending order

Tip: The first multiple of any number is the number itself.
Tip: Multiples continue infinitely, but we often list only the first few.
2 Multiple Identification
Exercise 2
Which of these numbers are multiples of 7? Circle them: 14, 18, 21, 25, 28, 32, 35, 42
Definition:

Multiple Test: To check if a number is a multiple of another, divide the first number by the second. If the result is a whole number, it's a multiple.

Base Number
7
Check Numbers
14, 18, 21, 25, 28, 32, 35, 42
Step 1: Test 14

14 ÷ 7 = 2 (whole number) → 14 is a multiple of 7

Step 2: Test 18

18 ÷ 7 = 2.57... (not whole) → 18 is not a multiple of 7

Step 3: Test 21

21 ÷ 7 = 3 (whole number) → 21 is a multiple of 7

Step 4: Test 25

25 ÷ 7 = 3.57... (not whole) → 25 is not a multiple of 7

Step 5: Test 28

28 ÷ 7 = 4 (whole number) → 28 is a multiple of 7

Step 6: Test 32

32 ÷ 7 = 4.57... (not whole) → 32 is not a multiple of 7

Step 7: Test 35

35 ÷ 7 = 5 (whole number) → 35 is a multiple of 7

Step 8: Test 42

42 ÷ 7 = 6 (whole number) → 42 is a multiple of 7

✓ 14
✗ 18
✓ 21
✗ 25
✓ 28
✗ 32
✓ 35
✓ 42
7 × 2 = 14
7 × 3 = 21
7 × 4 = 28
7 × 5 = 35
7 × 6 = 42
Multiples of 7 from the list: 14, 21, 28, 35, 42
Final answer:

The multiples of 7 from the given list are: 14, 21, 28, 35, 42

Applied rules:

• Division Test: A number is a multiple if division results in no remainder

• Verification: Multiply the quotient by the base number to confirm

• Systematic Testing: Test each number individually

3 Multiple Sequences
Exercise 3
Find the next three multiples of 9 after 36.
Definition:

Multiple Sequence: A series of multiples following a specific pattern.

Base Number
9
Starting Point
36
Step 1: Verify 36 is a multiple of 9

36 ÷ 9 = 4 (whole number) → 36 is a multiple of 9

Step 2: Find which multiple 36 represents

36 = 9 × 4, so 36 is the 4th multiple of 9

Step 3: Find the next three multiples

5th multiple: 9 × 5 = 45

6th multiple: 9 × 6 = 54

7th multiple: 9 × 7 = 63

Step 4: Verify each result

45 ÷ 9 = 5 ✓

54 ÷ 9 = 6 ✓

63 ÷ 9 = 7 ✓

45
54
63
9 × 5 = 45
9 × 6 = 54
9 × 7 = 63
Next three multiples of 9 after 36: 45, 54, 63
Final answer:

The next three multiples of 9 after 36 are: 45, 54, 63

Applied rules:

• Sequential Rule: Each multiple is the previous multiple plus the base number

• Verification Rule: Always divide to confirm the result

• Pattern Recognition: Multiples form arithmetic sequences

Rules and methods, laws,...
If n = a × b, then n is a multiple of both a and b
Multiple Definition
Multiples of 2
Even numbers
All even numbers
Multiples of 5
End in 0 or 5
Ending digit is 0 or 5
Multiples of 10
End in 0
Ending digit is 0
Key definitions:

Multiple: A number that can be divided by another number evenly

Factor: A number that divides another number evenly

Common Multiple: A number that is a multiple of two or more numbers

Least Common Multiple (LCM): The smallest multiple shared by two or more numbers

Complete methodology:
  1. Understand the concept: A multiple is formed by multiplying a number by whole numbers
  2. Systematic approach: Multiply by 1, 2, 3, 4... in order
  3. Pattern recognition: Each multiple is the previous plus the base number
  4. Verification: Divide to confirm the relationship
Tip 1: The first multiple of any number is the number itself.
Tip 2: Multiples are always greater than or equal to the base number.
Tip 3: Multiples form arithmetic sequences with a constant difference.
Tip 4: Every number is a multiple of itself and 1.
Common errors: Confusing multiples with factors, thinking multiples must be larger than the base number (they can be equal), forgetting that every number is a multiple of itself.
Key insight: The number of multiples is infinite for any non-zero number.
Multiple properties to know:

• Every number is a multiple of itself: n = n × 1

• Zero is a multiple of every number: 0 = n × 0

• Multiples are infinite: No largest multiple exists

• Relationship with factors: If a is a factor of b, then b is a multiple of a

• Arithmetic sequence: Multiples form sequences with constant differences

Solution: Exercises 4 to 5
4 True/False Multiples
Exercise 4
Determine if each statement is True or False:
a) 24 is a multiple of 6
b) 35 is a multiple of 8
c) 40 is a multiple of 5
d) 51 is a multiple of 9
Definition:

Multiple Test: A number is a multiple if it can be divided by another number with no remainder.

Statement A
24 ÷ 6 = 4
Result A
True
Step 1: Test statement a) 24 is a multiple of 6

24 ÷ 6 = 4 (whole number) → True

Step 2: Test statement b) 35 is a multiple of 8

35 ÷ 8 = 4.375 (not whole) → False

Step 3: Test statement c) 40 is a multiple of 5

40 ÷ 5 = 8 (whole number) → True

Step 4: Test statement d) 51 is a multiple of 9

51 ÷ 9 = 5.666... (not whole) → False

a) True
b) False
c) True
d) False
6 × 4 = 24 ✓
8 × ? ≠ 35 ✗
5 × 8 = 40 ✓
9 × ? ≠ 51 ✗
a) True, b) False, c) True, d) False
Final answer:

a) True, b) False, c) True, d) False

Applied rules:

• Division Test: Check if division results in a whole number

• Verification: Multiply to confirm the relationship

• Remainder Check: Multiples produce zero remainder

5 Common Multiples
Exercise 5
Find the first three common multiples of 4 and 6.
Definition:

Common Multiple: A number that is a multiple of two or more different numbers.

Number 1
4
Number 2
6
Common
Shared multiples
Step 1: List multiples of 4

4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48...

Step 2: List multiples of 6

6, 12, 18, 24, 30, 36, 42, 48, 54, 60...

Step 3: Identify common multiples

Compare the two lists: 12, 24, 36, 48...

First three: 12, 24, 36

Step 4: Verify each common multiple

12 ÷ 4 = 3 ✓ and 12 ÷ 6 = 2 ✓

24 ÷ 4 = 6 ✓ and 24 ÷ 6 = 4 ✓

36 ÷ 4 = 9 ✓ and 36 ÷ 6 = 6 ✓

Multiples of 4
4
8
12
16
20
24
28
32
36
Multiples of 6
6
12
18
24
30
36
Common Multiples
12
24
36
First three common multiples of 4 and 6: 12, 24, 36
Final answer:

The first three common multiples of 4 and 6 are: 12, 24, 36

Applied rules:

• Listing Method: Find multiples of each number separately, then compare

• Commonality: Shared multiples appear in both lists

• Verification: Check that each number divides the common multiple evenly

Complete Summary: Identifying Multiples

Key Definitions:

Multiple: A number that results from multiplying a given number by any whole number. For example, the multiples of 5 are 5, 10, 15, 20, 25, etc.

Base Number: The number whose multiples we are finding (e.g., 5 in the multiples of 5).

Multiplier: The whole number we multiply the base number by (e.g., 1, 2, 3, 4...).

Common Multiple: A number that is a multiple of two or more different numbers.

Least Common Multiple (LCM): The smallest number that is a multiple of two or more numbers.

Core Rules and Properties:

• Every number is a multiple of itself: n = n × 1

• Zero is a multiple of every number: 0 = n × 0

• Multiples are infinite: There is no largest multiple for any non-zero number

• Multiples are ≥ the base number: Multiples are greater than or equal to the base number

• Arithmetic sequence: Multiples form sequences with a constant difference equal to the base number

• Relationship with factors: If a is a factor of b, then b is a multiple of a

Step-by-Step Methods:
  1. Finding Multiples: Multiply the base number by 1, 2, 3, 4, etc.
  2. Identifying Multiples: Divide the candidate number by the base number; if the result is a whole number, it's a multiple
  3. Finding Common Multiples: List multiples of each number and identify shared values
  4. Verification: Always divide to confirm that the relationship holds true
Examples from Simple to Advanced:

Simple: Multiples of 3: 3, 6, 9, 12, 15...

Intermediate: Is 42 a multiple of 7? 42 ÷ 7 = 6, so yes.

Advanced: First common multiples of 4 and 6: 12, 24, 36...

Tip 1: Use skip counting to quickly identify multiples (count by 3s: 3, 6, 9...).
Tip 2: Multiples always end in predictable patterns based on the base number.
Tip 3: The LCM of two numbers is the smallest number that appears in both multiples lists.
Tip 4: Remember: factors divide into a number, multiples come from a number.
Common Mistakes: Confusing multiples with factors, thinking multiples must be larger than the base number (they can be equal), forgetting that every number is a multiple of itself.
Key Takeaway: Multiples form the foundation for understanding patterns, divisibility, and fraction operations.
Memory Aids:

• "Multiply to Get Multiples": Think "M" for Multiply, M for Multiples

• Counting Pattern: Multiples follow skip counting patterns

• Division Test: If division is exact, it's a multiple

• Visual Pattern: Multiples create evenly spaced numbers on a number line

Multiples and Patterns Comprehensive Guide
If n = a × b, then n is a multiple of both a and b
Multiple Definition
Key definitions:

Multiple: A number that can be divided by another number evenly without remainder

Factor: A number that divides another number evenly

Common Multiple: A number that is a multiple of two or more numbers

Least Common Multiple (LCM): The smallest multiple shared by multiple numbers

Arithmetic Sequence: A sequence with a constant difference between consecutive terms

Complete methodology:
  1. Systematic Approach: Multiply the base number by 1, 2, 3, 4... in order
  2. Pattern Recognition: Notice that each multiple is the previous plus the base number
  3. Verification: Divide the potential multiple by the base number to confirm
  4. Organization: List multiples in ascending order
Tip 1: Use divisibility rules to quickly identify potential multiples.
Tip 2: Multiples form arithmetic sequences with a constant difference.
Tip 3: The greatest common factor (GCF) relates to the pattern of common multiples.
Tip 4: Understanding multiples helps with fraction operations and reducing fractions.

Important Properties: Every number has infinitely many multiples, multiples are always ≥ the base number, and they form regular patterns.
Patterns: Multiples of 2 are even, multiples of 5 end in 0 or 5, multiples of 10 end in 0, etc.
Key relationships:

• Factor-Multiple Relationship: If a is a factor of b, then b is a multiple of a

• Commutative Property: If n is a multiple of a, and a is a multiple of b, then n is a multiple of b

• Closure Property: The sum of multiples of a number is also a multiple of that number

• LCM-GCD Connection: For any two numbers a and b: LCM(a,b) × GCD(a,b) = a × b

Questions & Answers

Question: I'm having trouble understanding how to find common multiples of two numbers. It seems like there are so many possibilities. How do I know which ones to pick?

Answer: Finding common multiples is like looking for shared interests between two groups! Here's how to approach it:

  1. List multiples of each number separately
  2. Look for numbers that appear in both lists
  3. These shared numbers are your common multiples

Example for 4 and 6:

  • Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40...
  • Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48...

Common multiples: 12, 24, 36...

The smallest one (12) is called the Least Common Multiple (LCM). Once you find the LCM, you can find other common multiples by adding the LCM repeatedly: 12, 24, 36, 48...

Question: What's the difference between multiples and factors? They seem similar but I know they're opposites somehow.

Answer: Great question! Multiples and factors are like two sides of the same coin:

Factors:

  • Numbers that divide into another number evenly
  • Smaller than or equal to the original number
  • Finite number of factors for any number
  • Example: Factors of 12 are 1, 2, 3, 4, 6, 12

Multiples:

  • Numbers you get when you multiply the original number
  • Larger than or equal to the original number
  • Infinite number of multiples
  • Example: Multiples of 12 are 12, 24, 36, 48, 60, ...

Think of it this way: factors go INTO the number (divide), multiples come FROM the number (multiply).

For number 8: Factors (go in) → 1, 2, 4, 8; Multiples (come from) → 8, 16, 24, 32, 40, ...

Question: My child is struggling with identifying patterns in multiples. Are there any visual aids or tricks that could help them see the patterns more clearly?

Answer: Yes, visual patterns are incredibly helpful for understanding multiples! Here are some effective strategies:

Visual Patterns:

  • Number Lines: Mark multiples on a number line to see their regular spacing
  • Hundred Charts: Color in multiples of a number to reveal diagonal or vertical patterns
  • Arrays: Arrange objects in rectangular arrays to show multiplication relationships

Pattern Recognition Tips:

  • Multiples of 2 always end in even digits (0, 2, 4, 6, 8)
  • Multiples of 5 always end in 0 or 5
  • Multiples of 10 always end in 0
  • Multiples of 3 have digits that add up to a multiple of 3

Memory Aids: Skip counting songs or chants can help internalize the patterns. Practice counting by 3s, 4s, 6s, etc., to build fluency with multiples.

The key is connecting the abstract concept to concrete visual representations!