Solved Exercises on Identifying Factors and Multiples in Grade 4

Master factors and multiples: finding factors, identifying multiples, and recognizing factor pairs through these 3 detailed exercises.

Solution: Exercises 1 to 3
1 Finding all factors
Exercise 1
Find all the factors of 24. List them in order from least to greatest.
Definition:

Factor: A number that divides another number evenly with no remainder

Factor pair: Two numbers that multiply together to make the target number

Systematic factor-finding method:
  1. Start with 1 and the number itself (1 × 24 = 24)
  2. Try 2: 2 × ? = 24 (2 × 12 = 24)
  3. Try 3: 3 × ? = 24 (3 × 8 = 24)
  4. Try 4: 4 × ? = 24 (4 × 6 = 24)
  5. Stop when factors repeat (5 doesn't divide 24 evenly)
Target
24
Factors
1, 2, 3, 4, 6, 8, 12, 24
Pairs
(1×24), (2×12), (3×8), (4×6)
24
1
24
2
12
3
8
4
6
Step 1: Start with 1 and the number

1 × 24 = 24, so 1 and 24 are factors

Step 2: Check 2

24 ÷ 2 = 12, so 2 and 12 are factors

Step 3: Check 3

24 ÷ 3 = 8, so 3 and 8 are factors

Step 4: Check 4

24 ÷ 4 = 6, so 4 and 6 are factors

Step 5: Check 5

24 ÷ 5 = 4 R4, so 5 is not a factor

Step 6: Stop when factors repeat

Since we already have 6, we stop here

1 × 24 = 24
2 × 12 = 24
3 × 8 = 24
4 × 6 = 24
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Final answer:

The factors of 24 are: 1, 2, 3, 4, 6, 8, 12, 24

We found them by systematically checking each number and seeing if it divides 24 evenly

Applied rules:

• Factor definition: A number that divides evenly into another number

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

• Systematic approach: Check numbers in order to avoid missing any

Tip 1: Factors always come in pairs that multiply to the target number.
Tip 2: Stop checking when you reach the square root or when factors repeat.
2 Identifying multiples
Exercise 2
List the first 5 multiples of 7. Then identify which of these are also multiples of 14.
Definition:

Multiple: The product of a number and any whole number

Example: Multiples of 7 are 7, 14, 21, 28, 35, 42, ...

Base
7
Multiples
7, 14, 21, 28, 35
Also of 14
14, 28
7 × 1 = 7
7 × 2 = 14
7 × 3 = 21
7 × 4 = 28
7 × 5 = 35
Step 1: Find multiples of 7

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

Step 2: Check which are also multiples of 14

• 7 ÷ 14 = 0 R7 (not a multiple of 14)

• 14 ÷ 14 = 1 (multiple of 14) ✓

• 21 ÷ 14 = 1 R7 (not a multiple of 14)

• 28 ÷ 14 = 2 (multiple of 14) ✓

• 35 ÷ 14 = 2 R7 (not a multiple of 14)

Step 3: Identify the pattern

Every second multiple of 7 is also a multiple of 14

This is because 14 = 2 × 7

Divisibility Check:

To verify if a number is a multiple of 7: Divide by 7 and check for remainder 0

To verify if a number is a multiple of 14: Divide by 14 and check for remainder 0

Multiples of 7: 7, 14, 21, 28, 35
Also multiples of 14: 14, 28
Final answer:

First 5 multiples of 7: 7, 14, 21, 28, 35

Multiples of both 7 and 14: 14, 28

These are the even multiples of 7 (since 14 = 2 × 7)

Applied rules:

• Multiple definition: n × whole number = multiple of n

• Pattern recognition: Every second multiple of 7 is a multiple of 14

• Verification: Check by division to confirm remainder is 0

Tip 1: Multiples form a counting sequence (skip counting).
Tip 2: If a is a multiple of b, then b is a factor of a.
3 Factor pairs and relationships
Exercise 3
Find all factor pairs of 36. Then explain the relationship between factors and multiples.
Definition:

Factor pair: Two numbers that multiply together to equal the target number

Relationship: If a is a factor of b, then b is a multiple of a

Target
36
Factors
1, 2, 3, 4, 6, 9, 12, 18, 36
Pairs
(1×36), (2×18), (3×12), (4×9), (6×6)
1
2
3
4
6
9
12
18
36
Step 1: Find factor pairs systematically

1 × 36 = 36, 2 × 18 = 36, 3 × 12 = 36, 4 × 9 = 36, 6 × 6 = 36

Step 2: List all factors in order

1, 2, 3, 4, 6, 9, 12, 18, 36

Step 3: Explain the factor-multiple relationship

If 4 is a factor of 36, then 36 is a multiple of 4

If 9 is a factor of 36, then 36 is a multiple of 9

Every factor-multiple relationship works both ways

1 × 36 = 36 (1 and 36 are factors)
2 × 18 = 36 (2 and 18 are factors)
3 × 12 = 36 (3 and 12 are factors)
4 × 9 = 36 (4 and 9 are factors)
6 × 6 = 36 (6 is a factor)
Factor-Multiple Relationship:
• If a divides b evenly, then a is a factor of b
• If b = a × n (where n is a whole number), then b is a multiple of a
• Example: 4 divides 36 (36 ÷ 4 = 9), so 4 is a factor of 36
• Example: 36 = 4 × 9, so 36 is a multiple of 4
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
Factor pairs: (1,36), (2,18), (3,12), (4,9), (6,6)
Final answer:

Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36

Factor pairs: (1,36), (2,18), (3,12), (4,9), (6,6)

Factors and multiples are inverse relationships: if a is a factor of b, then b is a multiple of a

Applied rules:

• Factor-multiple connection: Factors and multiples are opposite relationships

• Symmetric relationship: If a divides b, then b is divisible by a

• Completeness: Find all factor pairs to identify all factors

Tip 1: When you find one factor, you automatically find its pair.
Tip 2: Square numbers have one factor that pairs with itself (like 6 × 6 = 36).
Factors and Multiples Concepts
a × b = c
If a divides c evenly, then a is a factor of c
Factor
a divides c
a × ? = c
Multiple
c = a × n
c is multiple of a
Pair
a × b = c
(a,b) are factors
Key definitions:

Factor: A number that divides another number evenly with no remainder

Multiple: The product of a number and any whole number

Factor Pair: Two numbers that multiply together to make the target number

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

Composite Number: A number with more than two factors

Finding factors systematically:
  1. Start with 1 and the number itself
  2. Check each subsequent number in order
  3. For each factor found, record its pair
  4. Stop when factors begin to repeat
  5. List factors in ascending order
Factor Properties: Every number has at least two factors: 1 and itself
Multiple Properties: Every number has infinitely many multiples
Important relationships:

• Factor-multiple connection: If a is a factor of b, then b is a multiple of a

• Divisibility: a is a factor of b if b ÷ a has remainder 0

• Range: Factors are ≤ the number, multiples are ≥ the number

Tip 1: Factors come in pairs that multiply to the original number.
Tip 2: Stop checking for factors when you reach the square root.
Tip 3: Multiples form an infinite sequence (skip counting).
Tip 4: Prime numbers have exactly two factors.
Practice Problems and Relationships
48 = 6 × 8
6 and 8 are factors of 48
Example breakdown:

Problem: Find factors of 48

Step 1: 1 × 48 = 48 → Factors: 1, 48

Step 2: 2 × 24 = 48 → Factors: 2, 24

Step 3: 3 × 16 = 48 → Factors: 3, 16

Step 4: 4 × 12 = 48 → Factors: 4, 12

Step 5: 6 × 8 = 48 → Factors: 6, 8

Result: All factors: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48

Common factor-finding strategies:
  1. Division test: Divide the number by potential factors
  2. Multiplication pairs: Find two numbers that multiply to the target
  3. Factor trees: Break the number down into prime factors
  4. Divisibility rules: Use shortcuts for 2, 3, 5, etc.
Multiples Pattern:
Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60...
Multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80...
Common multiples: 24, 48 (LCM is smallest: 24)
Tip 1: Use divisibility rules to quickly identify potential factors.
Tip 2: A number is always both a factor and multiple of itself.
Tip 3: 1 is a factor of every number.
Verification methods:

• Multiplication check: Multiply factor pairs to verify they equal the original number

• Division check: Divide the original number by each factor to confirm remainder is 0

• Completeness: Ensure you've found all factor pairs

Practice 1
Find all factors of 30
Practice 2
List first 5 multiples of 9
Practice 3
Find all factor pairs of 20
Detailed Factors and Multiples Summary
a | b ↔ b = a × n
Factor Relationship
Core Concepts:

Factor: A number that divides another number evenly (no remainder)

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

Factor Pair: Two numbers that multiply together to equal a target number

Divisibility: When one number divides another with no remainder

Complete factor-finding procedure:
  1. Start systematically: Begin with 1 and the number itself
  2. Test sequentially: Check 2, 3, 4, etc. in order
  3. Record pairs: For each factor found, record its complement
  4. Stop appropriately: When factors begin to repeat or exceed the square root
  5. Organize results: List all factors in ascending order
Benefits: Builds number sense, prepares for fractions and algebra
Applications: Simplifying fractions, finding common denominators, solving equations
Fundamental rules:

• Factor range: Factors are always less than or equal to the number

• Multiple range: Multiples are always greater than or equal to the number

• Pair relationship: Factors always come in pairs that multiply to the original

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

Tip 1: Use divisibility rules to speed up factor identification.
Tip 2: When you find a factor, you automatically find its pair.
Tip 3: Stop checking when factors start repeating.
Tip 4: Square numbers have one factor that pairs with itself.
Common Mistakes to Avoid:

• Missing factors: Not checking systematically in order

• Repeating factors: Continuing past the square root unnecessarily

• Multiples confusion: Thinking multiples are limited in number

• Factor vs multiple: Confusing which is which in the relationship

Questions & Answers

Question: I sometimes confuse factors and multiples. How are they different?

Answer: Think of it this way:

  • Factors: Numbers that divide INTO another number evenly
  • Multiples: Numbers that result FROM multiplying the original number

Example with 12:
- Factors of 12: 1, 2, 3, 4, 6, 12 (these divide 12 evenly)
- Multiples of 12: 12, 24, 36, 48... (these are 12×1, 12×2, 12×3...)

Memory trick: "F" comes before "M" in the alphabet, just like factors are usually smaller than multiples!

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

Answer: You know you've found all factors when:

  • Factor pairs start repeating: You find (a,b) then later (b,a)
  • You reach the square root: When checking numbers up to √n
  • No new factors found: Testing numbers beyond the square root

For example, with 20: You check 1,2,3,4... When you reach 4, you get 4×5=20. Next would be 5×4=20 (already found). So you stop!

Question: How do factors and multiples connect to other math concepts?

Answer: Factors and multiples are foundational for:

  • Fractions: Simplifying fractions uses common factors
  • LCM/GCF: Finding least common multiples and greatest common factors
  • Algebra: Factoring polynomials uses similar principles
  • Patterns: Recognizing number sequences and relationships

Understanding factors and multiples provides the foundation for working with equivalent fractions, solving ratio problems, and eventually factoring quadratic equations in algebra.