Solved Exercises on Understanding Multiples in Grade 4

Master multiples: identifying multiples, finding common multiples, and multiple patterns through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Identifying Multiples
Exercise 1
Which of these numbers are multiples of 6: 12, 18, 23, 24, 30, 35, 36, 42? Explain your reasoning.
Definition:

Multiple: A number that can be expressed as the product of a given number and any whole number. If n × a = b, then b is a multiple of a.

Identifying Multiples Method:
  1. Divide each number: Divide by the given number (6 in this case)
  2. Check remainder: If remainder is 0, it's a multiple
  3. Verify with multiplication: Check that the number equals the given number times a whole number
Target Number
6
Test Numbers
12, 18, 23, 24, 30, 35, 36, 42
Multiples of 6
?
Step 1: Test each number for divisibility by 6

12 ÷ 6 = 2 R0 → 12 is a multiple of 6
18 ÷ 6 = 3 R0 → 18 is a multiple of 6
23 ÷ 6 = 3 R5 → 23 is NOT a multiple of 6
24 ÷ 6 = 4 R0 → 24 is a multiple of 6
30 ÷ 6 = 5 R0 → 30 is a multiple of 6
35 ÷ 6 = 5 R5 → 35 is NOT a multiple of 6
36 ÷ 6 = 6 R0 → 36 is a multiple of 6
42 ÷ 6 = 7 R0 → 42 is a multiple of 6

Step 2: Verify using multiplication

2 × 6 = 12 ✓
3 × 6 = 18 ✓
4 × 6 = 24 ✓
5 × 6 = 30 ✓
6 × 6 = 36 ✓
7 × 6 = 42 ✓

Step 3: List the multiples

The multiples of 6 from the given list are: 12, 18, 24, 30, 36, 42

Multiples of 6: 12, 18, 24, 30, 36, 42
Final answer:

The multiples of 6 are: 12, 18, 24, 30, 36, 42. These numbers can all be expressed as 6 multiplied by a whole number: 2×6=12, 3×6=18, 4×6=24, 5×6=30, 6×6=36, 7×6=42.

Applied rules:

• Multiple definition: A number is a multiple of another if it can be divided evenly by that number

• Divisibility test: A number is a multiple of 6 if it divides evenly with no remainder

• Verification: Always verify by expressing the multiple as the product of the base number and a whole number

2 Skip Counting Method
Exercise 2
Use skip counting to list the first 8 multiples of 7. Then explain how this method works.
Definition:

Skip counting: A method of finding multiples by repeatedly adding the same number to itself.

Base Number
7
Count
First 8
Multiples
?
Step 1: Start with the base number

First multiple: 7 × 1 = 7

Step 2: Keep adding the base number

Second multiple: 7 + 7 = 14
Third multiple: 14 + 7 = 21
Fourth multiple: 21 + 7 = 28
Fifth multiple: 28 + 7 = 35
Sixth multiple: 35 + 7 = 42
Seventh multiple: 42 + 7 = 49
Eighth multiple: 49 + 7 = 56

Step 3: Express as multiplication

1 × 7 = 7
2 × 7 = 14
3 × 7 = 21
4 × 7 = 28
5 × 7 = 35
6 × 7 = 42
7 × 7 = 49
8 × 7 = 56

Step 4: List the first 8 multiples

The first 8 multiples of 7 are: 7, 14, 21, 28, 35, 42, 49, 56

Step 5: Explain the method

Skip counting works because each multiple is the base number added to itself a specific number of times. The nth multiple is the base number multiplied by n.

First 8 multiples of 7: 7, 14, 21, 28, 35, 42, 49, 56
Final answer:

The first 8 multiples of 7 are: 7, 14, 21, 28, 35, 42, 49, 56. Skip counting works by repeatedly adding the base number to itself, which is equivalent to multiplying the base number by consecutive whole numbers.

Applied rules:

• Skip counting: Adding the same number repeatedly to generate multiples

• Multiplication connection: nth multiple = base number × n

• Pattern recognition: Multiples form arithmetic sequences with common difference equal to the base number

3 Multiples and Factors Relationship
Exercise 3
If 36 is a multiple of 4, what can you say about 4 in relation to 36? Explain the relationship between multiples and factors.
Definition:

Relationship between multiples and factors: If a number A is a multiple of number B, then B is a factor of A. These concepts are inversely related.

Given
36 is a multiple of 4
Therefore
4 is a factor of 36
Relationship
?
Step 1: Verify that 36 is a multiple of 4

36 ÷ 4 = 9 R0 → Yes, 36 is a multiple of 4
9 × 4 = 36 → Confirmed

Step 2: Apply the relationship

Since 36 = 9 × 4, this means 4 divides 36 evenly with no remainder.

Step 3: Define the relationship

If A is a multiple of B, then B is a factor of A.

Step 4: Apply to this example

Since 36 is a multiple of 4, then 4 is a factor of 36.

Step 5: Verify the factor relationship

36 ÷ 4 = 9 → 4 divides 36 evenly
4 × 9 = 36 → 4 and 9 are factor pairs of 36

Step 6: Generalize the relationship

Multiples and factors are inverse concepts: if a × b = c, then c is a multiple of both a and b, and both a and b are factors of c.

4 is a factor of 36
Final answer:

If 36 is a multiple of 4, then 4 is a factor of 36. The relationship between multiples and factors is inverse: when a number A is a multiple of number B, then B is a factor of A. In this case, 36 = 9 × 4, so 36 is a multiple of both 9 and 4, and both 9 and 4 are factors of 36.

Applied rules:

• Inverse relationship: If A is a multiple of B, then B is a factor of A

• Multiplication connection: a × b = c means c is a multiple of both a and b

• Division connection: If a × b = c, then c ÷ a = b and c ÷ b = a

Multiples: Definitions, Properties, and Methods
a × b = c, so c is a multiple of both a and b
Multiple Definition
Multiple Example
5 × 3 = 15
So 15 is a multiple of 5 and 3
Skip Counting
3, 6, 9, 12, 15...
Multiples of 3
Factor-Multiple
4 × 9 = 36
4 and 9 are factors of 36; 36 is multiple of both
Key definitions:

Multiple: A number that can be expressed as the product of a given number and any whole number.

Skip counting: A method of finding multiples by repeatedly adding the same number.

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

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

Finding multiples methods:
  1. Skip counting: Repeatedly add the base number to itself
  2. Multiplication: Multiply the base number by consecutive whole numbers
  3. Division test: Check if the number divides evenly by the base number
  4. Pattern recognition: Identify multiples based on known patterns
Tip 1: The first multiple of any number is the number itself (multiply by 1).
Tip 2: Multiples form arithmetic sequences with a common difference equal to the base number.
Tip 3: Every number is both a factor and a multiple of itself.
Tip 4: The multiples of a number are infinite - there is no largest multiple.
Multiple properties: Every number is a multiple of 1. Zero is a multiple of every number. Multiples are always greater than or equal to the base number (except for 0).
Factor-multiple relationship: If A is a multiple of B, then B is a factor of A. These concepts are inversely related.
Important multiples rules:

• Every number is a multiple of itself

• 0 is a multiple of every number

• Multiples are always greater than or equal to the base number (except for 0)

• If A is a multiple of B, then B is a factor of A

• Multiples form infinite arithmetic sequences

Multiple Patterns and Sequences
Pattern Recognition
Visualizing the patterns in multiples of different numbers.

Analysis: The chart shows how multiples form distinct patterns.

  • Multiples of 2 are even numbers
  • Multiples of 5 end in 0 or 5
  • Multiples of 10 end in 0
Solution: Exercises 4 to 5
4 Common Multiples
Exercise 4
Find the first 5 common multiples of 4 and 6. Explain how you found them.
Definition:

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

Numbers
4 and 6
Common Multiples
?
Method
?
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 lists to find numbers that appear in both:
12 (appears in both lists)
24 (appears in both lists)
36 (appears in both lists)
48 (appears in both lists)
60 (appears in both lists)

Step 4: Verify the common multiples

12 ÷ 4 = 3 R0 ✓ and 12 ÷ 6 = 2 R0 ✓
24 ÷ 4 = 6 R0 ✓ and 24 ÷ 6 = 4 R0 ✓
36 ÷ 4 = 9 R0 ✓ and 36 ÷ 6 = 6 R0 ✓
48 ÷ 4 = 12 R0 ✓ and 48 ÷ 6 = 8 R0 ✓
60 ÷ 4 = 15 R0 ✓ and 60 ÷ 6 = 10 R0 ✓

Step 5: Alternative method using LCM

First find LCM of 4 and 6:
Multiples of 4: 4, 8, 12, 16, 20, 24...
Multiples of 6: 6, 12, 18, 24, 30...
LCM(4,6) = 12
Other common multiples: 12×2=24, 12×3=36, 12×4=48, 12×5=60

First 5 common multiples: 12, 24, 36, 48, 60
Final answer:

The first 5 common multiples of 4 and 6 are: 12, 24, 36, 48, 60. These numbers appear in both the list of multiples of 4 and the list of multiples of 6. The LCM of 4 and 6 is 12, and all other common multiples are multiples of 12.

Applied rules:

• Common multiple definition: A number that is a multiple of all given numbers

• LCM method: All common multiples are multiples of the LCM

• Verification: Check that each common multiple divides evenly by all original numbers

5 Multiple Patterns and Applications
Exercise 5
A bus arrives every 8 minutes and a train arrives every 12 minutes. If both arrive at the station at 9:00 AM, when will they next arrive at the same time? Explain your reasoning using multiples.
Definition:

Real-world applications: Problems involving scheduling, timing, and periodic events often involve finding common multiples, specifically the Least Common Multiple (LCM).

Bus Interval
8 minutes
Train Interval
12 minutes
Next Same Time
?
Step 1: Understand the problem

We need to find when both the bus and train will arrive simultaneously after 9:00 AM. This occurs at a time that is a multiple of both 8 and 12 minutes.

Step 2: Find multiples of each interval

Multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80...
Multiples of 12: 12, 24, 36, 48, 60, 72, 84, 96...

Step 3: Identify the first common multiple

Looking for the smallest number that appears in both lists: 24

Step 4: Verify 24 is a common multiple

24 ÷ 8 = 3 R0 ✓
24 ÷ 12 = 2 R0 ✓

Step 5: Calculate the arrival time

Starting time: 9:00 AM
Minutes until next simultaneous arrival: 24 minutes
9:00 AM + 24 minutes = 9:24 AM

Step 6: Explain the reasoning

The bus will arrive at 9:08, 9:16, 9:24...
The train will arrive at 9:12, 9:24...
The first time they both arrive together is at 9:24 AM.

Next simultaneous arrival: 9:24 AM
Final answer:

The bus and train will next arrive at the same time at 9:24 AM. This occurs because 24 is the least common multiple of 8 and 12, meaning it's the first time that is a multiple of both intervals. After 24 minutes from 9:00 AM, both the bus (arriving every 8 minutes) and the train (arriving every 12 minutes) will arrive simultaneously.

Applied rules:

• LCM application: For periodic events, the LCM gives the time until next simultaneous occurrence

• Real-world connection: Common multiples apply to scheduling and timing problems

• Verification: Check that the solution satisfies the conditions of the problem

Complete Summary: Understanding Multiples
a × b = c, so c is a multiple of both a and b
Multiple Definition
Key definitions:

Multiple: A number that can be expressed as the product of a given number and any whole number. If a × b = c, then c is a multiple of both a and b.

Skip counting: A method of finding multiples by repeatedly adding the same number to itself.

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

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

Factor-multiple relationship: If A is a multiple of B, then B is a factor of A.

Complete methodology:
  1. Identify the base number: Know what number you're finding multiples of.
  2. Choose a method: Skip counting, multiplication, or division test.
  3. Generate multiples: Use your chosen method to list multiples.
  4. For common multiples: List multiples of each number and identify overlaps.
  5. Find LCM if needed: Identify the smallest common multiple.
  6. Verify your results: Ensure each multiple divides evenly by the base number(s).
Tip 1: The first multiple of any number is the number itself (multiply by 1).
Tip 2: Multiples form arithmetic sequences with a common difference equal to the base number.
Tip 3: To find common multiples, it's often efficient to find the LCM first.
Tip 4: Remember the inverse relationship: if A is a multiple of B, then B is a factor of A.
Common errors: Confusing multiples with factors, forgetting that every number is a multiple of itself, stopping too early when looking for common multiples.
Key patterns: Multiples of 2 are even, multiples of 5 end in 0 or 5, multiples of 10 end in 0, multiples form infinite arithmetic sequences.
Multiple rules:

• Every number is a multiple of itself

• 0 is a multiple of every number

• Multiples are always greater than or equal to the base number (except for 0)

• If A is a multiple of B, then B is a factor of A

• Multiples form infinite arithmetic sequences

• All common multiples of two numbers are multiples of their LCM

Exercise with Visualization: Multiple Relationships
Exercise 6: LCM Visualization
Visualizing how multiples of different numbers relate:
f₁(x) = Multiples of 4 (for x = 1-10)
f₂(x) = Multiples of 6 (for x = 1-10)
f₃(x) = Common multiples of 4 and 6 (for x = 1-10)

Analysis: The chart shows how multiples of different numbers intersect.

  • Common multiples occur at regular intervals
  • The LCM is the first intersection point
  • Common multiples form their own arithmetic sequence

Questions & Answers

Question: I'm confused about the difference between multiples and factors. Can you explain?

Answer: Think of multiples and factors as opposites:

  • Multiples: Numbers you get BY multiplying. For 3: 3, 6, 9, 12, 15...
  • Factors: Numbers you multiply TOGETHER to get a number. Factors of 12: 1, 2, 3, 4, 6, 12

Remember: "Multiples are many" (infinite), "Factors are few" (limited).

Question: How do I know if a number is a multiple of another number? Sometimes I make mistakes.

Answer: Use the division test:

  • Divide the potential multiple by the base number
  • If the remainder is 0, it's a multiple
  • If there's a remainder, it's not a multiple

Example: Is 24 a multiple of 6? 24 ÷ 6 = 4 R0 → Yes, 24 is a multiple of 6.

Question: My child is struggling with skip counting to find multiples. Any tips to make it easier?

Answer: Try these strategies:

  • Use visual aids: Count objects like blocks or coins
  • Use a number line: Hop along the line by the base number
  • Start small: Begin with multiples of 2 and 5 before moving to others
  • Practice rhythmically: Say multiples in a rhythmic pattern
  • Connect to multiplication: Show that skip counting is repeated addition, which is multiplication

Consistent practice will build fluency!

Question: Why do we need to learn about common multiples? When will I use this in real life?

Answer: Common multiples have many real-world applications:

  • Scheduling: When do two recurring events happen at the same time?
  • Transportation: When do two buses/train schedules align?
  • Manufacturing: When do two production cycles synchronize?
  • Planning: Finding when multiple activities will coincide

Like in the exercise: finding when bus and train arrive together!

Question: Are there patterns in multiples that can help me identify them quickly?

Answer: Yes! Here are some helpful patterns:

  • Multiples of 2: Always even (end in 0, 2, 4, 6, 8)
  • Multiples of 5: End in 0 or 5
  • Multiples of 10: Always end in 0
  • Multiples of 3: Sum of digits is a multiple of 3
  • Multiples of 9: Sum of digits is a multiple of 9

These patterns can help you quickly identify multiples!