Solved Exercises on Finding Multiples in Grade 4

Master finding multiples: definition, identification, patterns, and applications through these 5 detailed exercises.

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

Multiple: A number that can be divided evenly by another number. Multiples of 3 are: 3×1, 3×2, 3×3, etc.

Method:
  1. Multiply the base number by 1, 2, 3, 4, 5, 6...
  2. List the results in order
  3. Check that each result is divisible by the base number
Base Number
3
Multipliers
1,2,3,4,5,6
Multiples
3,6,9,12,15,18
Step 1: Multiply by 1

3 × 1 = 3

Step 2: Multiply by 2

3 × 2 = 6

Step 3: Multiply by 3

3 × 3 = 9

Step 4: Multiply by 4

3 × 4 = 12

Step 5: Multiply by 5

3 × 5 = 15

Step 6: Multiply by 6

3 × 6 = 18

3
6
9
12
15
18
↓ Pattern: Add 3 each time ↓
The first 6 multiples of 3 are: 3, 6, 9, 12, 15, 18
Final answer:

3, 6, 9, 12, 15, 18

Applied rules:

• Multiplication: Base number × whole number = multiple

• Pattern: Each multiple is the previous + base number

• Division: Every multiple ÷ base number = whole number

2 Identifying Multiples
Exercise 2
Which of these numbers are multiples of 4? 8, 12, 15, 16, 20, 22, 24
Definition:

Multiples of 4: Numbers that can be expressed as 4 × n where n is a whole number.

Numbers to Check
8,12,15,16,20,22,24
Divide by 4
✓,✓,✗,✓,✓,✗,✓
Multiples
8,12,16,20,24
Step 1: Test each number

8 ÷ 4 = 2 ✓ (whole number)

12 ÷ 4 = 3 ✓ (whole number)

15 ÷ 4 = 3.75 ✗ (not whole)

16 ÷ 4 = 4 ✓ (whole number)

20 ÷ 4 = 5 ✓ (whole number)

22 ÷ 4 = 5.5 ✗ (not whole)

24 ÷ 4 = 6 ✓ (whole number)

Step 2: Identify pattern

Multiples of 4 end in: 0, 2, 4, 6, 8 (even digits)

Last two digits form a number divisible by 4

Step 3: List multiples

8, 12, 16, 20, 24

8 ✓
12 ✓
15 ✗
16 ✓
20 ✓
22 ✗
24 ✓
Multiples of 4: 8, 12, 16, 20, 24
Final answer:

8, 12, 16, 20, 24 are multiples of 4

Applied rules:

• Division test: Multiple ÷ base = whole number

• Pattern: Multiples of 4 are always even

• Quick check: Last 2 digits divisible by 4

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

Common Multiple: A number that is a multiple 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 multiples of 3

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

Step 2: List multiples of 4

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

Step 3: Find common numbers

Compare lists: 12 appears in both

Next: 24 appears in both

Then: 36 appears in both

Step 4: Verify

12 ÷ 3 = 4 ✓ and 12 ÷ 4 = 3 ✓

24 ÷ 3 = 8 ✓ and 24 ÷ 4 = 6 ✓

36 ÷ 3 = 12 ✓ and 36 ÷ 4 = 9 ✓

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
First 3 common multiples: 12, 24, 36
Final answer:

12, 24, 36

Applied rules:

• Listing method: Write multiples of each number

• Comparison: Find numbers appearing in both lists

• Verification: Check divisibility by both numbers

Rules and methods, laws,...
Multiple = Base × Whole\ Number
Definition of Multiple
Rule 1
n × 1 = n
Any number is a multiple of itself
Rule 2
n × 0 = 0
0 is a multiple of every number
Rule 3
n × m = mn
Multiplication creates multiples
Key definitions:

Multiple: Product of a number and any integer

Factor: Number that divides another number evenly

Common Multiple: Multiple shared by two or more numbers

Complete methodology:
  1. Identify the base number: Determine which number's multiples you need
  2. Apply multiplication: Multiply by 1, 2, 3, 4, etc.
  3. Check divisibility: Verify each result ÷ base = whole number
  4. Find patterns: Look for sequences and relationships
Tip 1: Multiples always increase by the base number.
Tip 2: Even numbers have only even multiples.
Tip 3: Use multiplication tables to find multiples quickly.
Tip 4: The smallest common multiple is called LCM.
Patterns to remember: Multiples of 5 end in 0 or 5, multiples of 10 end in 0.
Key insight: Every number has infinite multiples, starting from itself.
Formulas to know by heart:

• Multiple of n = n × k (where k is any whole number)

• First multiple of any number = the number itself

• Zero is a multiple of every number

• Multiples form arithmetic sequences

Solution: Exercises 4 to 5
4 Finding Missing Multiples
Exercise 4
Complete the sequence: 7, 14, __, 28, __, 42
Definition:

Arithmetic Sequence: A sequence where each term increases by a constant amount (the common difference).

Given
7,14,__,28,__,42
Pattern
Add 7
Complete
7,14,21,28,35,42
Step 1: Identify the pattern

14 - 7 = 7, so the pattern is adding 7

This means these are multiples of 7

Step 2: Find missing numbers

After 14: 14 + 7 = 21

After 28: 28 + 7 = 35

Step 3: Verify

7×1=7, 7×2=14, 7×3=21, 7×4=28, 7×5=35, 7×6=42

7×1=7
7×2=14
7×3=21
7×4=28
7×5=35
7×6=42
Groups of 7
Complete sequence: 7, 14, 21, 28, 35, 42
Final answer:

Missing numbers: 21, 35

Applied rules:

• Pattern recognition: Find the common difference

• Sequential counting: Add the same number repeatedly

• Multiplication: Each term is base × position

5 Real-world Application
Exercise 5
Sarah packs cookies in boxes of 6. She needs at least 50 cookies. What is the smallest number of cookies she can pack?
Definition:

Real-world application: Using multiples to solve practical problems involving grouping and packaging.

Box size
6
Minimum needed
50
Smallest multiple ≥ 50
54
Step 1: List multiples of 6

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

Step 2: Find the first multiple ≥ 50

48 < 50 (too small)

54 ≥ 50 (this works!)

Step 3: Verify

54 ÷ 6 = 9 boxes (whole number)

54 ≥ 50 ✓

No smaller multiple of 6 satisfies the condition

Step 4: Answer the question

Sarah needs 54 cookies (9 boxes of 6)

Boxes: 1
Cookies: 6
Boxes: 2
Cookies: 12
Boxes: 3
Cookies: 18
Boxes: 4
Cookies: 24
Boxes: 5
Cookies: 30
Boxes: 6
Cookies: 36
Boxes: 7
Cookies: 42
Boxes: 8
Cookies: 48
Boxes: 9
Cookies: 54
Smallest number of cookies: 54
Final answer:

Sarah needs 54 cookies (9 full boxes)

Applied rules:

• Problem solving: Translate words to mathematical operations

• Minimum requirement: Find smallest multiple ≥ target

• Practical application: Connect math to real situations

Multiples Properties and Patterns
Multiple = Factor × Whole\ Number
Multiples Formula
Key definitions:

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

Factor: A number that divides another number evenly

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

Finding Multiples Methods:
  1. Multiplication method: Multiply base number by 1, 2, 3, 4...
  2. Addition method: Start with base number, keep adding it
  3. Division test: Divide candidate by base, check if result is whole
  4. Pattern recognition: Use known patterns (ending digits, etc.)
Tip 1: Multiples of 2 are always even numbers.
Tip 2: Multiples of 5 end in 0 or 5.
Tip 3: Multiples of 10 always end in 0.
Tip 4: Sum of digits of multiples of 3 is also a multiple of 3.
Tip 5: Every number is a multiple of its factors.
Multiples patterns: They form arithmetic sequences with common difference equal to the base number.
Number relationships: If A is a multiple of B, then B is a factor of A.
Important Properties:

• Every number is a multiple of 1

• Every number is a multiple of itself

• 0 is a multiple of every number

• Multiples are always ≥ the base number (except 0)

• There are infinitely many multiples of any number

Multiples Patterns Infographic
📊
Multiples of 2
Pattern: Even numbers
2
4
6
8
10
Multiples of 3
Pattern: Sum of digits ÷ 3
3
6
9
12
15
Multiples of 5
Pattern: End in 0 or 5
5
10
15
20
25
How to Find Multiples Step-by-Step
1
Choose Base Number
2
Multiply by 1,2,3...
3
List Results
4
Verify Pattern
Multiples Relationships
Every multiple of 6 is also a multiple of 2 and 3
Common multiples of 2 and 3: 6, 12, 18, 24...
Remember: Multiples Always Increase!

Questions & Answers

Question: I don't understand how to find multiples quickly. Is there an easy way to remember them?

Answer: Yes! Here are some quick strategies:

  • Counting by the number: For multiples of 4, count: 4, 8, 12, 16, 20...
  • Use multiplication tables: If you know 4×3=12, then 12 is a multiple of 4
  • Look for patterns: Multiples of 5 always end in 0 or 5
  • Add the base number: Once you have one multiple, add the base number to get the next

For example, multiples of 7: Start with 7, then 7+7=14, then 14+7=21, then 21+7=28, etc.

Practice with small numbers first, then work up to larger ones!

Question: How do I know if a big number like 144 is a multiple of 12?

Answer: There are several ways to check:

  • Division test: 144 ÷ 12 = 12 (whole number, so yes!)
  • Factor breakdown: 12 = 3×4, so 144 must be divisible by both 3 and 4
  • Pattern check: Since 12 = 3×4, check if 144 follows both patterns

For 144: It ends in 44, and 44÷4=11 (so divisible by 4). Sum of digits: 1+4+4=9, and 9÷3=3 (so divisible by 3). Since it's divisible by both 3 and 4, it's divisible by 12.

Therefore, 144 is a multiple of 12: 12×12=144.

Question: What's the difference between factors and multiples? I always mix them up.

Answer: Great question! Here's the difference:

  • Factors: Numbers that divide into another number evenly
  • Multiples: Numbers that result from multiplying a number by another

Example with 12:

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

Think of it this way: factors go INTO a number, multiples come FROM a number.

Memory trick: "F" for factors goes first alphabetically, just like factors are usually smaller numbers. "M" for multiples comes later, just like multiples are usually bigger numbers.