Solved Exercises on Finding Multiples of Numbers in Grade 4

Master finding multiples: definition, methods, properties, and practical applications through these 5 detailed exercises.

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

Multiple: A multiple of a number is the product of that number and any whole number. For example, multiples of 6 are 6×1, 6×2, 6×3, etc.

Method for finding multiples:
  1. Multiply the given number by 1, 2, 3, 4, 5, etc.
  2. List the results in order
  3. Each result is a multiple of the original number
Number
6
Multiplication
6×1, 6×2, 6×3, 6×4, 6×5
Results
6, 12, 18, 24, 30
Step 1: Multiply by 1

6 × 1 = 6

Step 2: Multiply by 2

6 × 2 = 12

Step 3: Multiply by 3

6 × 3 = 18

Step 4: Multiply by 4

6 × 4 = 24

Step 5: Multiply by 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: Multiply the base number by counting numbers (1, 2, 3, 4, 5)

• Pattern Recognition: Multiples increase by the base number each time

• Order: List multiples in ascending order

2 Identifying multiples
Exercise 2
Which numbers from the list are multiples of 7?
14, 21, 25, 28, 33, 35, 42, 49
Definition:

Identifying multiples: A number is a multiple of another if it can be divided evenly (with no remainder) by that number.

List
14, 21, 25, 28, 33, 35, 42, 49
Divide by 7
÷7
Multiples
14, 21, 28, 35, 42, 49
Step 1: Check each number

14 ÷ 7 = 2 (remainder 0) ✓

21 ÷ 7 = 3 (remainder 0) ✓

25 ÷ 7 = 3 R4 (remainder 4) ✗

28 ÷ 7 = 4 (remainder 0) ✓

33 ÷ 7 = 4 R5 (remainder 5) ✗

35 ÷ 7 = 5 (remainder 0) ✓

42 ÷ 7 = 6 (remainder 0) ✓

49 ÷ 7 = 7 (remainder 0) ✓

Step 2: List multiples of 7

From the division checks above, multiples of 7 are: 14, 21, 28, 35, 42, 49

Step 3: Identify non-multiples

Numbers that are NOT multiples of 7: 25, 33

Multiples of 7: 14, 21, 28, 35, 42, 49
Final answer:

The multiples of 7 from the list are: 14, 21, 28, 35, 42, 49

Applied rules:

• Division Test: Divide each number by the base number and check for remainder

• No Remainder Rule: If division has no remainder, it's a multiple

• Verification: Check that each identified multiple can be expressed as base number × whole number

3 Listing multiples up to a limit
Exercise 3
List all multiples of 8 that are less than 50.
Definition:

Multiples up to a limit: Continue multiplying the base number by consecutive whole numbers until the result exceeds the specified limit.

Base Number
8
Limit
< 50
Result
8, 16, 24, 32, 40, 48
Step 1: Start with first multiple

8 × 1 = 8 (less than 50) ✓

Step 2: Continue multiplying

8 × 2 = 16 (less than 50) ✓

8 × 3 = 24 (less than 50) ✓

8 × 4 = 32 (less than 50) ✓

8 × 5 = 40 (less than 50) ✓

8 × 6 = 48 (less than 50) ✓

8 × 7 = 56 (greater than 50) ✗

Step 3: Stop when limit exceeded

Stop at 8 × 6 = 48 because 8 × 7 = 56 > 50

Multiples of 8 less than 50: 8, 16, 24, 32, 40, 48
Final answer:

All multiples of 8 less than 50 are: 8, 16, 24, 32, 40, 48

Applied rules:

• Limit Rule: Stop generating multiples once you exceed the specified limit

• Continuity: List multiples in ascending order without skipping

• Boundary Check: Verify that the last multiple is indeed less than the limit

Key Concepts: Multiples, Properties, and Methods
Multiple of n = n × k (where k is any whole number)
Definition of Multiples
Property 1
Every number is a multiple of itself
5 is a multiple of 5 (5×1=5)
Property 2
Every number is a multiple of 1
7 is a multiple of 1 (7×1=7)
Property 3
Zero is a multiple of every number
0 is a multiple of 3 (3×0=0)
Key definitions:

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

Factors: Numbers that divide another number evenly

Divisibility: When one number divides another with no remainder

Complete methodology:
  1. Understand the problem: Identify if you need to find, list, or identify multiples
  2. Apply the definition: Use multiplication (base number × whole number)
  3. Follow constraints: Respect limits or ranges if specified
  4. Verify your answer: Check that each result fits the criteria
Tip 1: Multiples form an arithmetic sequence with the base number as the common difference.
Tip 2: The smallest positive multiple of any number is the number itself.
Tip 3: To quickly check if a number is a multiple, divide and see if remainder is 0.
Tip 4: Multiples of 10 always end in 0; multiples of 5 end in 0 or 5.
Common errors: Including numbers that aren't multiples, forgetting to start from 0 or 1, miscounting.
Memory aids: Think of multiples as "skip counting" - count by the base number.
Essential facts about multiples:

• Every number has infinitely many multiples

• The multiples of a number form an infinite sequence

• If a is a multiple of b, then b is a factor of a

• Multiples of even numbers are always even

• Multiples of 5 end in 0 or 5

Multiples Visualization: Patterns and Relationships
Exercise 6: Comparing Multiples
Compare the multiples of 4 and 6 to find common multiples:
Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40...
Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48...

Analysis: The chart shows the relationship between multiples of different numbers.

  • Common multiples of 4 and 6: 12, 24, 36, 48...
  • The smallest common multiple is called the LCM (Least Common Multiple)
  • LCM of 4 and 6 is 12

Questions & Answers

Question: I don't understand how multiples relate to multiplication. Are they the same thing?

Answer: Great question! Multiples and multiplication are closely related but not exactly the same:

  • Multiplication: The operation of combining equal groups (e.g., 3 × 4 = 12)
  • Multiples: The results of multiplication when you multiply a specific number by counting numbers

Think of it this way: if you're learning multiples of 5, you're really just doing the 5 times table: 5×1=5, 5×2=10, 5×3=15, 5×4=20, etc. These results (5, 10, 15, 20...) are the multiples of 5.

So multiples ARE the answers to multiplication problems where one factor stays the same and the other increases by 1 each time.

Question: Why is zero considered a multiple of every number? That seems strange.

Answer: Zero is a multiple of every number because of the definition of multiplication:

  • Any number multiplied by 0 equals 0: n × 0 = 0
  • Since multiples are formed by multiplying a number by whole numbers (0, 1, 2, 3...), zero is always included
  • For example: 5 × 0 = 0, so 0 is a multiple of 5

This might seem counterintuitive, but it's mathematically consistent. It helps make mathematical rules work properly. For instance, if you're listing all multiples of 3 under 15, you'd include: 0, 3, 6, 9, 12.

However, in many practical problems, teachers focus on positive multiples (1, 2, 3, 4...) because they're more intuitive for young learners.

Question: How can I quickly tell if a big number is a multiple of 3 without dividing?

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

  • 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 432 a multiple of 3? 4+3+2 = 9, and 9 is a multiple of 3, so YES! - Is 578 a multiple of 3? 5+7+8 = 20, and 20 is not a multiple of 3, so NO!

This works because of how our base-10 number system relates to the number 3. It's a quick way to check without doing long division!

Similar tricks exist: for 9 (add digits, check if sum is multiple of 9), for 5 (ends in 0 or 5), for 10 (ends in 0).

Question: What's the difference between factors and multiples? They seem similar.

Answer: Factors and multiples are opposite concepts, like "parts" and "wholes":

  • Factors: Numbers that divide evenly into another number (the parts that make up a number)
  • Multiples: Numbers that result from multiplying a number by whole numbers (the results of a number)

Using the number 12 as an example: - Factors of 12: 1, 2, 3, 4, 6, 12 (these divide 12 evenly) - Multiples of 12: 12, 24, 36, 48, 60... (results of 12×1, 12×2, 12×3...)

Think of it like this: If A × B = C, then A and B are factors of C, and C is a multiple of both A and B.

In simpler terms: factors are smaller (or equal), multiples are larger (or equal).

Question: How do multiples help me solve real-world problems?

Answer: Multiples are everywhere in daily life! Here are some practical applications:

  1. Time: Minutes come in multiples of 5 (5, 10, 15, 20...) on analog clocks
  2. Money: Counting coins uses multiples (nickels = multiples of 5, dimes = multiples of 10)
  3. Grouping: If you have 6 boxes with 4 items each, you have multiples of 4 (4, 8, 12, 16, 20, 24)
  4. Patterns: Understanding multiples helps with calendar patterns, schedules, and sequences

For example: If a bus comes every 15 minutes, the times it arrives are multiples of 15: 15, 30, 45, 60 minutes past the hour.

Multiples also prepare you for more advanced math like finding common denominators, least common multiples, and greatest common factors.