Solved Exercises on Multiples in Grade 4

Master multiples: finding multiples, identifying common multiples, and solving word problems through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Finding multiples
Exercise 1
List the first 5 multiples of 4.
Definition:

Multiple: A number that results from multiplying another number by an integer (1, 2, 3, ...)

Method:
  1. Multiply the base number by 1, 2, 3, 4, 5, ...
  2. Continue until you have the required number of multiples
  3. List the results in order
Base Number
4
Multiplication
× 1, 2, 3, 4, 5
Result
4, 8, 12, 16, 20
Step 1: Multiply by 1

4 × 1 = 4

Step 2: Multiply by 2

4 × 2 = 8

Step 3: Multiply by 3

4 × 3 = 12

Step 4: Multiply by 4

4 × 4 = 16

Step 5: Multiply by 5

4 × 5 = 20

The first 5 multiples of 4 are: 4, 8, 12, 16, 20
Final answer:

The first 5 multiples of 4 are: 4, 8, 12, 16, 20

Applied rules:

• Multiplication: A multiple of n is n × any positive integer

• Order: Multiples are listed in ascending order

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

2 Identifying multiples
Exercise 2
Which of these numbers are multiples of 6? 12, 18, 24, 27, 30, 35
Definition:

Multiple check: A number is a multiple of another if it divides evenly (no remainder)

Numbers
12, 18, 24, 27, 30, 35
Division by 6
÷ 6
Results
2, 3, 4, 4.5, 5, 5.83...
Step 1: Check each number

12 ÷ 6 = 2 ✓ (whole number)

18 ÷ 6 = 3 ✓ (whole number)

24 ÷ 6 = 4 ✓ (whole number)

27 ÷ 6 = 4.5 ✗ (not whole)

30 ÷ 6 = 5 ✓ (whole number)

35 ÷ 6 = 5.83... ✗ (not whole)

Step 2: Identify multiples

Numbers that divide evenly by 6 are multiples of 6

Step 3: List the multiples

12, 18, 24, 30

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

The multiples of 6 are: 12, 18, 24, 30

Applied rules:

• Division test: If division has no remainder, it's a multiple

• Whole number: Only whole number quotients indicate multiples

• Pattern recognition: Multiples increase by the base number

3 Common multiples
Exercise 3
Find the first 3 common multiples of 4 and 6.
Definition:

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

Numbers
4 and 6
Multiples of 4
4, 8, 12, 16, 20, 24...
Multiples of 6
6, 12, 18, 24, 30, 36...
Common
12, 24, 36
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: Find common numbers

Compare both lists: 12, 24, 36 appear in both lists

Step 4: List the first 3 common multiples

12, 24, 36

First 3 common multiples of 4 and 6: 12, 24, 36
Final answer:

The first 3 common multiples of 4 and 6 are: 12, 24, 36

Applied rules:

• Listing method: Write out multiples of each number

• Comparison: Look for numbers appearing in both lists

• LCM connection: The smallest common multiple is the LCM

Multiples: Rules and Methods
Multiple of n = n × k (where k = 1, 2, 3, ...)
Multiple Formula
Definition
Multiple
Result of multiplying a number by an integer
Common Multiple
LCM
Smallest positive number divisible by both
Test
Division
If division is exact, it's a multiple
Key definitions:

Multiple: The product of a given number and any whole number

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

Least Common Multiple (LCM): The smallest positive common multiple

Complete methodology:
  1. Find multiples: Multiply the base number by 1, 2, 3, etc.
  2. Identify multiples: Check if a number divides evenly by the base
  3. Find common multiples: List multiples of each number and find overlaps
  4. Verify results: Check calculations by reversing operations
Tip 1: Multiples form arithmetic sequences (add the same number each time).
Tip 2: Every number is a multiple of itself and 1.
Tip 3: Zero is a multiple of every number.
Tip 4: Use multiplication tables to quickly identify multiples.
Common errors: Confusing factors with multiples, forgetting zero is a multiple, misidentifying common multiples.
Memory aid: "Multiples make bigger numbers" (except zero).
Important rules:

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

• Every number has infinitely many multiples

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

• The smallest positive multiple of any number is the number itself

• Common multiples are found by listing and comparing

Solution: Exercises 4 to 5
4 Word problem
Exercise 4
Sarah practices piano every 3 days and swimming every 4 days. If she did both activities today, in how many days will she do both again?
Definition:

Word problem: Real-world situation requiring mathematical solution

Activities
Piano: every 3 days, Swimming: every 4 days
Find
Common multiple
Answer
12 days
Step 1: Identify the pattern

Piano: Day 0, 3, 6, 9, 12, 15, 18, 21, 24...

Swimming: Day 0, 4, 8, 12, 16, 20, 24, 28...

Step 2: Find common multiples

Both occur on day 0, 12, 24, 36...

Step 3: Find the next occurrence

After day 0, the next common day is day 12

Step 4: Verify the result

Day 12: Piano (12÷3=4, whole number) ✓

Day 12: Swimming (12÷4=3, whole number) ✓

Sarah will do both activities again in 12 days
Final answer:

Sarah will practice piano and swim again in 12 days

Applied rules:

• Real-world context: Translate words into mathematical relationships

• LCM concept: Find least common multiple for simultaneous events

• Verification: Check that the answer fits the problem conditions

5 Pattern recognition
Exercise 5
The first four multiples of 7 are 7, 14, 21, 28. What are the next three multiples? Describe the pattern.
Definition:

Pattern recognition: Identifying regularities in sequences of numbers

Given
7, 14, 21, 28
Pattern
Add 7 each time
Next 3
35, 42, 49
Step 1: Observe the pattern

7, 14, 21, 28 - each number increases by 7

Step 2: Continue the pattern

28 + 7 = 35

35 + 7 = 42

42 + 7 = 49

Step 3: Verify using multiplication

7 × 5 = 35 ✓

7 × 6 = 42 ✓

7 × 7 = 49 ✓

Step 4: Describe the pattern

All multiples of 7 increase by 7 from one to the next

Next 3 multiples: 35, 42, 49. Pattern: Add 7 to get the next multiple.
Final answer:

The next three multiples of 7 are: 35, 42, 49

The pattern is that each multiple is 7 more than the previous one.

Applied rules:

• Arithmetic sequence: Multiples form sequences with constant differences

• Pattern continuation: Extend sequences by following the established rule

• Verification: Confirm new terms using multiplication

Multiples: Laws, Methods, Definitions, and Examples
Multiple of n = n × k (where k = 1, 2, 3, ...)
Multiple Formula
Key definitions:

Multiple: A number that results from multiplying another number by a whole 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

Complete methodology:
  1. Find multiples: Multiply the base number by 1, 2, 3, 4, 5, ...
  2. Identify multiples: Check if a number divides evenly by the base number
  3. Find common multiples: List multiples of each number and identify overlaps
  4. Solve word problems: Recognize that simultaneous events often involve LCM
  5. Verify results: Always check your answers by reversing the operation
Tip 1: Multiples of a number form an arithmetic sequence with a common difference equal to the number itself.
Tip 2: To check if a number is a multiple of another, divide and see if there's no remainder.
Tip 3: The smallest positive multiple of any number is the number itself.
Tip 4: When finding common multiples, start with the larger number's multiples to find the LCM faster.
Tip 5: In word problems, simultaneous recurring events often require finding the LCM.
Important laws: Every number is a multiple of itself; zero is a multiple of every number; every number has infinitely many multiples.
Common mistakes: Confusing multiples with factors; forgetting that 0 is a multiple of every number; not recognizing that multiples continue infinitely.
Key formulas and rules:

• Multiple of n = n × k (where k is any positive integer)

• To check if A is a multiple of B: A ÷ B should result in a whole number

• Common multiples of A and B are numbers divisible by both A and B

• The Least Common Multiple (LCM) is the smallest positive common multiple

• Multiples of any number form an infinite arithmetic sequence

Multiples Visualization: Patterns and Relationships
Exercise 6: Multiples Patterns
Compare the multiples of 3 and 4 up to 24:
Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24
Multiples of 4: 4, 8, 12, 16, 20, 24
Find their common multiples and LCM.

Analysis: The chart shows how multiples of different numbers create distinct patterns.

  • Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24 (increase by 3)
  • Multiples of 4: 4, 8, 12, 16, 20, 24 (increase by 4)
  • Common multiples: 12, 24
  • Least Common Multiple (LCM): 12

Questions & Answers

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

Answer: Great question! Think of factors and multiples as opposites:

  • Factors: Numbers that divide into another number evenly (go inside the number)
  • Multiples: Numbers that result from multiplying a number (come after the number)

For example, with the number 12:

  • Factors of 12: 1, 2, 3, 4, 6, 12 (these divide into 12)
  • Multiples of 12: 12, 24, 36, 48, 60... (these come after 12 when multiplying)

Memory trick: "F" comes before "M" in the alphabet, just like factors are smaller numbers that fit into a number, while multiples are larger numbers that come after.

Question: Why is zero considered a multiple of every number? That doesn't make sense to me.

Answer: This is a great observation! Zero is a multiple of every number because of the definition of multiples:

  • A multiple of a number n is n × k where k is any whole number
  • If k = 0, then n × 0 = 0
  • This works for ANY number n, so 0 is a multiple of every number

Examples:

  • 0 = 5 × 0, so 0 is a multiple of 5
  • 0 = 12 × 0, so 0 is a multiple of 12
  • 0 = 100 × 0, so 0 is a multiple of 100

However, in most practical problems, we focus on positive multiples (1, 2, 3, ...) unless specifically asked about zero.

Question: Is there a faster way to find common multiples without listing all of them?

Answer: Yes, there are some shortcuts for finding common multiples, especially the LCM (Least Common Multiple):

  • For small numbers: List multiples of the larger number and see which ones the smaller number also divides evenly
  • Look for patterns: If one number is a multiple of the other, the LCM is the larger number
  • Use prime factorization: Break numbers into prime factors and take the highest power of each

Example: For 6 and 9

  • Multiples of 9: 9, 18, 27, 36...
  • Does 6 divide these? 9÷6=1.5 (no), 18÷6=3 (yes!) → LCM is 18

For now, listing multiples is the most reliable method for grade 4 students. As you advance, you'll learn more sophisticated methods!