Solved Exercises on How to Use Multiples in Grade 4

Master how to use multiples: finding multiples, identifying patterns, solving word problems, and understanding divisibility relationships through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Finding Multiples
Exercise 1
List the first 5 multiples of 6.
Definition:

Multiples: Numbers that result from multiplying a given number by whole numbers (0, 1, 2, 3, ...)

Method:
  1. Multiply the number by 0, 1, 2, 3, 4, ...
  2. Continue until you have the required number of multiples
  3. Each multiple is evenly divisible by the original number
Number
6
Multiplication
× 1, 2, 3, 4, 5
Result
6, 12, 18, 24, 30
Step 1: Start with 6 × 1

6 × 1 = 6

Step 2: Continue with 6 × 2

6 × 2 = 12

Step 3: Continue with 6 × 3

6 × 3 = 18

Step 4: Continue with 6 × 4

6 × 4 = 24

Step 5: Continue with 6 × 5

6 × 5 = 30

First 5 multiples of 6: 6, 12, 18, 24, 30
Final answer:

The first 5 multiples of 6 are: 6, 12, 18, 24, 30

Applied rules:

• Multiplication: Any number multiplied by whole numbers gives its multiples

• Pattern: Multiples increase by the original number each time

• Divisibility: Each multiple is divisible by the original number

2 Identifying Multiples
Exercise 2
Which numbers from the list are multiples of 4?
8, 15, 16, 22, 24, 30, 32
Definition:

Multiples of 4: Numbers that can be divided evenly by 4 (remainder = 0)

Test each number
÷ 4
Results
No remainder?
Multiples
8, 16, 24, 32
Step 1: Check 8 ÷ 4

8 ÷ 4 = 2 remainder 0 → YES, 8 is a multiple of 4

Step 2: Check 15 ÷ 4

15 ÷ 4 = 3 remainder 3 → NO, 15 is not a multiple of 4

Step 3: Check 16 ÷ 4

16 ÷ 4 = 4 remainder 0 → YES, 16 is a multiple of 4

Step 4: Check 22 ÷ 4

22 ÷ 4 = 5 remainder 2 → NO, 22 is not a multiple of 4

Step 5: Check 24 ÷ 4

24 ÷ 4 = 6 remainder 0 → YES, 24 is a multiple of 4

Step 6: Check 30 ÷ 4

30 ÷ 4 = 7 remainder 2 → NO, 30 is not a multiple of 4

Step 7: Check 32 ÷ 4

32 ÷ 4 = 8 remainder 0 → YES, 32 is a multiple of 4

Multiples of 4: 8, 16, 24, 32
Final answer:

The multiples of 4 from the list are: 8, 16, 24, 32

Applied rules:

• Division Test: A number is a multiple of another if division has no remainder

• Multiplication Check: Can the number be expressed as the original number × whole number?

• Pattern Recognition: Multiples of 4 end in specific digits (0, 2, 4, 6, 8)

3 Pattern Recognition
Exercise 3
Find the next 3 multiples of 7 after 21.
Definition:

Pattern of multiples: Each multiple increases by the original number

Current sequence
7, 14, 21
Next 3
28, 35, 42
Add 7 each time
+7, +7, +7
Step 1: Identify the pattern

7, 14, 21... Each number increases by 7

Step 2: Add 7 to 21

21 + 7 = 28

Step 3: Add 7 to 28

28 + 7 = 35

Step 4: Add 7 to 35

35 + 7 = 42

Step 5: Verify the results

28 ÷ 7 = 4, 35 ÷ 7 = 5, 42 ÷ 7 = 6 → All divide evenly

Next 3 multiples of 7 after 21: 28, 35, 42
Final answer:

The next 3 multiples of 7 after 21 are: 28, 35, 42

Applied rules:

• Constant Addition: Multiples increase by the original number each time

• Pattern Recognition: Look for consistent intervals between numbers

• Verification: Always check that new numbers divide evenly by the original

Rules and methods, laws,...
Multiple = Number × Whole Number
Definition of Multiples
Rule 1
Multiples increase by the original number
Pattern Rule
Rule 2
Multiples divide evenly by the original
Division Rule
Rule 3
Every number is a multiple of itself
Identity Rule
Key definitions:

Multiples: Products of a number and whole numbers (0, 1, 2, 3...)

Factors: Numbers that divide evenly into another number

Patterns: Consistent sequences showing relationships between numbers

Complete methodology:
  1. Identify the base number: Determine which number you're finding multiples of
  2. Choose the operation: Multiply by whole numbers (0, 1, 2, 3...)
  3. Apply the method: Follow the chosen technique consistently
  4. Verify the result: Check by dividing or recognizing the pattern
Tip 1: The first multiple of any number is the number itself.
Tip 2: Zero is a multiple of every number (any number × 0 = 0).
Tip 3: Multiples always end in predictable patterns based on the original number.
Tip 4: Always verify by dividing the supposed multiple by the original number.
Common errors: Confusing multiples with factors, forgetting zero as a multiple, miscounting when listing multiples.
Key concepts: Multiples continue infinitely, every number is a multiple of its factors, multiples form arithmetic sequences.
Rules to know by heart:

• Every number is a multiple of 1 and itself

• Zero is a multiple of every number

• 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 Word Problem
Exercise 4
Sarah buys pencils in packs of 8. She needs at least 50 pencils for her class. What is the smallest number of packs she should buy?
Definition:

Word problems: Real-world situations requiring mathematical solutions using multiples

Pack size
8 pencils
Requirement
≥ 50 pencils
Solution
7 packs = 56 pencils
Step 1: List multiples of 8

8, 16, 24, 32, 40, 48, 56, 64, ...

Step 2: Find the first multiple ≥ 50

48 < 50, but 56 ≥ 50

Step 3: Determine number of packs

56 ÷ 8 = 7 packs

Step 4: Verify the answer

6 packs = 48 pencils (not enough), 7 packs = 56 pencils (sufficient)

Sarah needs 7 packs (56 pencils)
Final answer:

Sarah should buy 7 packs to get at least 50 pencils.

Applied rules:

• Real-world application: Connect multiples to practical situations

• Inequality solving: Find the smallest multiple meeting requirements

• Verification: Check that the solution meets all conditions

5 Common Multiples
Exercise 5
Find the first 3 common multiples of 3 and 4.
Definition:

Common multiples: Numbers that are multiples of two or more numbers

Multiples of 3
3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36
Multiples of 4
4, 8, 12, 16, 20, 24, 28, 32, 36, 40
Common
12, 24, 36
Step 1: List first several multiples of 3

3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36

Step 2: List first several multiples of 4

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

Step 3: Identify common numbers

12 appears in both lists, 24 appears in both lists, 36 appears in both lists

Step 4: Verify each common multiple

12 ÷ 3 = 4, 12 ÷ 4 = 3 → Both divide evenly

24 ÷ 3 = 8, 24 ÷ 4 = 6 → Both divide evenly

36 ÷ 3 = 12, 36 ÷ 4 = 9 → Both divide evenly

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

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

Applied rules:

• Systematic listing: Organize multiples in order to identify patterns

• Comparison: Compare two sets of multiples to find common elements

• Verification: Confirm that common multiples divide by both original numbers

Key Concepts: Multiples, Patterns, and Applications
Multiple = Base Number × Count
Formula for Multiples
Key definitions:

Multiples: Numbers obtained by multiplying a base number by whole numbers (0, 1, 2, 3...)

Common Multiples: Numbers that are multiples of two or more different numbers

Patterns: Consistent sequences showing mathematical relationships

Complete methodology:
  1. Identify the base number: Determine which number you're working with
  2. Choose the operation: Decide whether to find multiples or identify them
  3. Apply the method: Follow systematic steps for the chosen technique
  4. Verify the result: Check answers using division or pattern recognition
Tip 1: Multiples of 2 are always even numbers (end in 0, 2, 4, 6, 8).
Tip 2: Multiples of 5 always end in 0 or 5.
Tip 3: To find common multiples, list multiples of each number and find overlaps.
Tip 4: The smallest common multiple is called the Least Common Multiple (LCM).
Important facts: Every number is a multiple of 1, zero is a multiple of every number, multiples continue infinitely.
Real-world applications: Scheduling events, packaging items, organizing groups, finding common denominators.
Key rules and patterns:

• Multiples of 10 always end in 0

• Multiples of 3 have digit sums that are multiples of 3

• Multiples of 9 have digit sums that are multiples of 9

• Multiples of 6 are multiples of both 2 and 3

• The set of multiples forms an infinite arithmetic sequence

Questions & Answers

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

Answer: Great question! Here's the key difference:

  • Multiples: These are the "answers" you get when you multiply a number by 1, 2, 3, etc.
  • Factors: These are the numbers that divide evenly into another number.

For example, with the number 12:

  • Multiples of 12: 12, 24, 36, 48, 60... (12×1, 12×2, 12×3...)
  • Factors of 12: 1, 2, 3, 4, 6, 12 (numbers that divide 12 evenly)

Think of it this way: multiples are the "products" of multiplication, while factors are the "divisors" that go into a number evenly.

Question: How do I quickly know if a number is a multiple of 3 without dividing it?

Answer: There's a helpful trick for multiples of 3!

Rule: Add up all the digits in the number. If the sum is a multiple of 3, then the original number is also a multiple of 3.

Examples:

  • Is 246 a multiple of 3? 2 + 4 + 6 = 12. Since 12 is a multiple of 3, 246 is a multiple of 3.
  • Is 178 a multiple of 3? 1 + 7 + 8 = 16. Since 16 is not a multiple of 3, 178 is not a multiple of 3.

This works because of how our base-10 number system relates to the number 3!

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

Answer: Multiples have many practical applications in daily life!

  • Shopping: Buying items sold in packages (packs of 6, boxes of 12) - calculating totals
  • Cooking: Scaling recipes up or down using multiples
  • Events: Planning schedules that repeat at regular intervals
  • Organizing: Arranging people or objects in equal groups
  • Time: Understanding how minutes relate to hours (multiples of 60)

For example, if you're planning a party and plates come in packages of 8, knowing multiples helps you figure out how many packages you need for 30 guests.