Solved Exercises on How to Do Multiples and Factors in Grade 4

Master how to do multiples and factors: definition, identification, and patterns through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Finding Multiples
Exercise 1
List the first 5 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...
  2. List the results in order
  3. Check that each result is divisible by the base number
3 × 1 = 3
3 × 2 = 6
3 × 3 = 9
3 × 4 = 12
3 × 5 = 15
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

3
6
9
12
15
Pattern: Add 3 each time
First 5 multiples of 3: 3, 6, 9, 12, 15
Final answer:

3, 6, 9, 12, 15

Applied rules:

• Multiplication: Base number × whole number = multiple

• Pattern: Each multiple is the previous + base number

• Division: Every multiple ÷ base number = whole number

2 Finding Factors
Exercise 2
Find all factors of 12.
Definition:

Factor: A number that divides another number evenly without remainder.

1 × 12 = 12
2 × 6 = 12
3 × 4 = 12
Step 1: Find factor pairs

1 × 12 = 12

2 × 6 = 12

3 × 4 = 12

Step 2: List all factors

Factors of 12: 1, 2, 3, 4, 6, 12

Step 3: Verify each factor

12 ÷ 1 = 12 ✓

12 ÷ 2 = 6 ✓

12 ÷ 3 = 4 ✓

12 ÷ 4 = 3 ✓

12 ÷ 6 = 2 ✓

12 ÷ 12 = 1 ✓

1
2
3
4
6
12
All factors divide 12 evenly
Factors of 12: 1, 2, 3, 4, 6, 12
Final answer:

1, 2, 3, 4, 6, 12

Applied rules:

• Factor pairs: Numbers that multiply to give the original number

• Division test: Factor ÷ original = whole number

• Completeness: All factors come in pairs

3 Comparing Multiples and Factors
Exercise 3
Find multiples of 4 up to 20 and factors of 20.
Definition:

Multiples vs Factors: Multiples are bigger numbers made by multiplying, factors are smaller numbers that divide evenly.

Multiples of 4: 4, 8, 12, 16, 20
Factors of 20: 1, 2, 4, 5, 10, 20
Step 1: Find multiples of 4

4×1=4, 4×2=8, 4×3=12, 4×4=16, 4×5=20

Step 2: Find factors of 20

Factor pairs: 1×20, 2×10, 4×5

Factors: 1, 2, 4, 5, 10, 20

Step 3: Compare the lists

Multiples of 4: 4, 8, 12, 16, 20

Factors of 20: 1, 2, 4, 5, 10, 20

Step 4: Notice the relationship

4 and 20 appear in both lists

This shows the connection between multiples and factors

Multiples: 4,8,12,16,20
Factors: 1,2,4,5,10,20
4 is both a multiple of 4 and a factor of 20
Multiples of 4: 4, 8, 12, 16, 20 | Factors of 20: 1, 2, 4, 5, 10, 20
Final answer:

Multiples of 4: 4, 8, 12, 16, 20 | Factors of 20: 1, 2, 4, 5, 10, 20

Applied rules:

• Multiples: Numbers made by multiplying the base

• Factors: Numbers that divide the original evenly

• Relationship: The same number can be both a multiple and factor

Multiples and Factors Rules
Multiple = Base × Whole\ Number
Multiple Definition
Rule 1
n × 1 = n
Any number is a multiple of itself
Rule 2
n ÷ n = 1
Any number is a factor of itself
Rule 3
n ÷ 1 = n
1 is a factor of every number
Key definitions:

Multiple: A number that can be divided evenly by another number

Factor: A number that divides another number evenly without remainder

Factor Pair: Two numbers that multiply to give the original number

Prime Number: A number with exactly 2 factors: 1 and itself

Complete methodology:
  1. For multiples: Multiply the base number by 1, 2, 3, 4...
  2. For factors: Find all pairs of numbers that multiply to give the original
  3. Verification: Check that multiples ÷ base = whole number, factors divide evenly
  4. Organization: List in ascending order
Tip 1: Multiples always increase by the base number.
Tip 2: Factors come in pairs that multiply to the original number.
Tip 3: 1 and the number itself are always factors.
Tip 4: Use multiplication tables to find multiples quickly.
Patterns to remember: Multiples of 5 end in 0 or 5, multiples of 10 end in 0.
Key insight: Every number has infinite multiples but finite factors.
Multiples and Factors Rules:

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

• Every number has infinite multiples

• Every number has a finite number of factors

• 1 is a factor of every number

• Every number is a factor of itself

Solution: Exercises 4 to 5
4 Common Multiples
Exercise 4
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
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...

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 ✓

Common: 12
Common: 24
Common: 36
12 is the Least Common Multiple (LCM)
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

5 Prime Factor Identification
Exercise 5
Find factors of 18 and identify which are prime numbers.
Definition:

Prime Factor: A factor that is also a prime number (has exactly 2 factors).

1 × 18 = 18
2 × 9 = 18
3 × 6 = 18
Step 1: Find factor pairs

1 × 18 = 18

2 × 9 = 18

3 × 6 = 18

Step 2: List all factors

Factors of 18: 1, 2, 3, 6, 9, 18

Step 3: Identify prime numbers

Prime numbers have exactly 2 factors: 1 and themselves

2: factors are 1, 2 → PRIME ✓

3: factors are 1, 3 → PRIME ✓

Others are composite

Step 4: Verify prime factors

2 is prime and divides 18: 18 ÷ 2 = 9 ✓

3 is prime and divides 18: 18 ÷ 3 = 6 ✓

1
2 (PRIME)
3 (PRIME)
6
9
18
Prime factors: 2 and 3
Factors: 1, 2, 3, 6, 9, 18 | Prime factors: 2, 3
Final answer:

Factors of 18: 1, 2, 3, 6, 9, 18 | Prime factors: 2, 3

Applied rules:

• Prime identification: Numbers with exactly 2 factors

• Prime factor: A factor that is also prime

• Verification: Confirm divisibility and primality

Multiples and Factors Complete Guide
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

Prime Number: A number with exactly two factors: 1 and itself

Finding Multiples and Factors Methods:
  1. Multiples method: Multiply base number by 1, 2, 3, 4...
  2. Factors method: Find all pairs of numbers that multiply to give the target
  3. Division test: Divide candidate by target to check if result is whole
  4. Verification: Ensure all factors divide evenly and multiples are exact
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

• Every number has at least 2 factors: 1 and itself

Multiples and Factors Patterns
📊
Multiples of 2
Pattern: Even numbers
2
4
6
8
10
Multiples of 3
Pattern: Sum of digits ÷ 3
3
6
9
12
15
Factors of 12
Pairs that multiply to 12
1×12
2×6
3×4
How to Find Multiples and Factors
1
Choose number
2
Find multiples
3
Find factors
4
Verify
Multiples vs Factors Relationship
Every factor of 12: 1,2,3,4,6,12
Multiples of 12: 12,24,36,48...
Connection: 12 is both factor and multiple
Remember: Multiples Get Bigger, Factors Get Smaller!

Questions & Answers

Question: How do I know if I've found all the factors of a number?

Answer: Here's how to be sure you've found all factors:

  • Check systematically: Go through numbers 1, 2, 3, 4... in order
  • Stop at the middle: When you reach the square root, you've found all pairs
  • Look for pairs: Every factor has a partner that multiplies to the original number

For example, with 12: Check 1 (works: 1×12), 2 (works: 2×6), 3 (works: 3×4), 4 (already paired with 3), stop here. Factors: 1, 2, 3, 4, 6, 12.

Once you start seeing repeated factors in your pairs, you're done!

Question: How do I know if a number is a multiple of another number?

Answer: There are several ways to check:

  • Division test: Divide the number by the potential factor
  • Remainder check: If the division has no remainder, it's a multiple
  • Multiplication check: See if you can multiply the base number to get the target

For example, is 24 a multiple of 6? 24 ÷ 6 = 4 (no remainder), so yes! Or 6×4=24, so yes!

If there's a remainder when dividing, it's not a multiple.

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.