Using Factors to Multiply - Practice Exercises

Master multiplication using factors: find factor pairs, use factor relationships, and apply multiplication strategies with these 10 detailed exercises.

Exercises 1 to 5
1 Factor Pairs for 12
Exercise 1
List all factor pairs of 12 and use them to multiply.
Definition:

Factor pairs: Two numbers that multiply together to give the original number.

Finding factor pairs method:
  1. Start with 1 and check if it divides evenly into the number
  2. Continue with 2, 3, 4, etc., up to the square root of the number
  3. For each divisor found, pair it with the quotient
Factor pairs
1×12, 2×6, 3×4
Products
12
All factors
1, 2, 3, 4, 6, 12
Step 1: Start with 1

12 ÷ 1 = 12, so 1×12 = 12

Step 2: Try 2

12 ÷ 2 = 6, so 2×6 = 12

Step 3: Try 3

12 ÷ 3 = 4, so 3×4 = 12

Step 4: Stop at square root

√12 ≈ 3.46, so we stop at 3

Step 5: List all factors

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

Factor pairs of 12: (1,12), (2,6), (3,4)
Final answer:

Factor pairs of 12: (1,12), (2,6), (3,4)

Applied rules:

• Factor definition: A number that divides another number evenly

• Pair relationship: Factor × Factor = Original number

• Systematic approach: Check divisors up to √n

2 Factor Pairs for 18
Exercise 2
List all factor pairs of 18 and use them to multiply.
Definition:

Factor pairs: Two numbers that multiply together to give the original number.

Factor pairs
1×18, 2×9, 3×6
Products
18
All factors
1, 2, 3, 6, 9, 18
Step 1: Start with 1

18 ÷ 1 = 18, so 1×18 = 18

Step 2: Try 2

18 ÷ 2 = 9, so 2×9 = 18

Step 3: Try 3

18 ÷ 3 = 6, so 3×6 = 18

Step 4: Stop at square root

√18 ≈ 4.24, so we stop at 4

Step 5: Check 4

18 ÷ 4 = 4.5, not a factor

Factor pairs of 18: (1,18), (2,9), (3,6)
Final answer:

Factor pairs of 18: (1,18), (2,9), (3,6)

Applied rules:

• Factor definition: A number that divides another number evenly

• Pair relationship: Factor × Factor = Original number

• Systematic approach: Check divisors up to √n

3 Using Factor Relationships: 15×4
Exercise 3
Use factor relationships to multiply 15×4.
Definition:

Factor relationships: Using known factor pairs to solve multiplication problems.

Factor 15
15 = 3×5
Rewrite
(3×5)×4
Regroup
3×(5×4)
Step 1: Break apart 15 into factors

15 = 3×5 (since 3×5 = 15)

Step 2: Rewrite the multiplication

15×4 = (3×5)×4

Step 3: Use associative property

(3×5)×4 = 3×(5×4)

Step 4: Calculate inner multiplication

5×4 = 20

Step 5: Complete the multiplication

3×20 = 60

15×4 = 60
Final answer:

15×4 = 60

Applied rules:

• Factorization: Breaking a number into its factors

• Associative property: (a×b)×c = a×(b×c)

• Multiplication strategy: Use known facts to solve unknown facts

4 Factor Pairs for 24
Exercise 4
List all factor pairs of 24 and use them to multiply.
Definition:

Factor pairs: Two numbers that multiply together to give the original number.

Factor pairs
1×24, 2×12, 3×8, 4×6
Products
24
All factors
1, 2, 3, 4, 6, 8, 12, 24
Step 1: Start with 1

24 ÷ 1 = 24, so 1×24 = 24

Step 2: Try 2

24 ÷ 2 = 12, so 2×12 = 24

Step 3: Try 3

24 ÷ 3 = 8, so 3×8 = 24

Step 4: Try 4

24 ÷ 4 = 6, so 4×6 = 24

Step 5: Stop at square root

√24 ≈ 4.9, so we stop at 4

Factor pairs of 24: (1,24), (2,12), (3,8), (4,6)
Final answer:

Factor pairs of 24: (1,24), (2,12), (3,8), (4,6)

Applied rules:

• Factor definition: A number that divides another number evenly

• Pair relationship: Factor × Factor = Original number

• Systematic approach: Check divisors up to √n

5 Using Factor Relationships: 14×6
Exercise 5
Use factor relationships to multiply 14×6.
Definition:

Factor relationships: Using known factor pairs to solve multiplication problems.

Factor 14
14 = 2×7
Rewrite
(2×7)×6
Regroup
2×(7×6)
Step 1: Break apart 14 into factors

14 = 2×7 (since 2×7 = 14)

Step 2: Rewrite the multiplication

14×6 = (2×7)×6

Step 3: Use associative property

(2×7)×6 = 2×(7×6)

Step 4: Calculate inner multiplication

7×6 = 42

Step 5: Complete the multiplication

2×42 = 84

14×6 = 84
Final answer:

14×6 = 84

Applied rules:

• Factorization: Breaking a number into its factors

• Associative property: (a×b)×c = a×(b×c)

• Multiplication strategy: Use known facts to solve unknown facts

Exercises 6 to 10
6 Factor Pairs for 30
Exercise 6
List all factor pairs of 30 and use them to multiply.
Definition:

Factor pairs: Two numbers that multiply together to give the original number.

Factor pairs
1×30, 2×15, 3×10, 5×6
Products
30
All factors
1, 2, 3, 5, 6, 10, 15, 30
Step 1: Start with 1

30 ÷ 1 = 30, so 1×30 = 30

Step 2: Try 2

30 ÷ 2 = 15, so 2×15 = 30

Step 3: Try 3

30 ÷ 3 = 10, so 3×10 = 30

Step 4: Try 5

30 ÷ 5 = 6, so 5×6 = 30

Step 5: Stop at square root

√30 ≈ 5.48, so we stop at 5

Factor pairs of 30: (1,30), (2,15), (3,10), (5,6)
Final answer:

Factor pairs of 30: (1,30), (2,15), (3,10), (5,6)

Applied rules:

• Factor definition: A number that divides another number evenly

• Pair relationship: Factor × Factor = Original number

• Systematic approach: Check divisors up to √n

7 Using Factor Relationships: 18×5
Exercise 7
Use factor relationships to multiply 18×5.
Definition:

Factor relationships: Using known factor pairs to solve multiplication problems.

Factor 18
18 = 2×9
Rewrite
(2×9)×5
Regroup
2×(9×5)
Step 1: Break apart 18 into factors

18 = 2×9 (since 2×9 = 18)

Step 2: Rewrite the multiplication

18×5 = (2×9)×5

Step 3: Use associative property

(2×9)×5 = 2×(9×5)

Step 4: Calculate inner multiplication

9×5 = 45

Step 5: Complete the multiplication

2×45 = 90

18×5 = 90
Final answer:

18×5 = 90

Applied rules:

• Factorization: Breaking a number into its factors

• Associative property: (a×b)×c = a×(b×c)

• Multiplication strategy: Use known facts to solve unknown facts

8 Factor Pairs for 36
Exercise 8
List all factor pairs of 36 and use them to multiply.
Definition:

Factor pairs: Two numbers that multiply together to give the original number.

Factor pairs
1×36, 2×18, 3×12, 4×9, 6×6
Products
36
All factors
1, 2, 3, 4, 6, 9, 12, 18, 36
Step 1: Start with 1

36 ÷ 1 = 36, so 1×36 = 36

Step 2: Try 2

36 ÷ 2 = 18, so 2×18 = 36

Step 3: Try 3

36 ÷ 3 = 12, so 3×12 = 36

Step 4: Try 4

36 ÷ 4 = 9, so 4×9 = 36

Step 5: Try 6

36 ÷ 6 = 6, so 6×6 = 36

Step 6: Stop at square root

√36 = 6, so we stop at 6

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

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

Applied rules:

• Factor definition: A number that divides another number evenly

• Pair relationship: Factor × Factor = Original number

• Systematic approach: Check divisors up to √n

9 Using Factor Relationships: 21×4
Exercise 9
Use factor relationships to multiply 21×4.
Definition:

Factor relationships: Using known factor pairs to solve multiplication problems.

Factor 21
21 = 3×7
Rewrite
(3×7)×4
Regroup
3×(7×4)
Step 1: Break apart 21 into factors

21 = 3×7 (since 3×7 = 21)

Step 2: Rewrite the multiplication

21×4 = (3×7)×4

Step 3: Use associative property

(3×7)×4 = 3×(7×4)

Step 4: Calculate inner multiplication

7×4 = 28

Step 5: Complete the multiplication

3×28 = 84

21×4 = 84
Final answer:

21×4 = 84

Applied rules:

• Factorization: Breaking a number into its factors

• Associative property: (a×b)×c = a×(b×c)

• Multiplication strategy: Use known facts to solve unknown facts

10 Using Factor Relationships: 16×7
Exercise 10
Use factor relationships to multiply 16×7.
Definition:

Factor relationships: Using known factor pairs to solve multiplication problems.

Factor 16
16 = 4×4
Rewrite
(4×4)×7
Regroup
4×(4×7)
Step 1: Break apart 16 into factors

16 = 4×4 (since 4×4 = 16)

Step 2: Rewrite the multiplication

16×7 = (4×4)×7

Step 3: Use associative property

(4×4)×7 = 4×(4×7)

Step 4: Calculate inner multiplication

4×7 = 28

Step 5: Complete the multiplication

4×28 = 112

16×7 = 112
Final answer:

16×7 = 112

Applied rules:

• Factorization: Breaking a number into its factors

• Associative property: (a×b)×c = a×(b×c)

• Multiplication strategy: Use known facts to solve unknown facts

Complete Summary: Using Factors to Multiply

Factor Concepts 🧮

📊
Step 1: Identify Factors
Find numbers that divide evenly into the target number
Step 2: List Factor Pairs
Write as pairs that multiply to the target number
Step 3: Use Factor Relationships
Break apart numbers to make multiplication easier
Step 4: Apply Properties
Use associative property to regroup multiplication
Step 5: Verify Product
Check that factors multiply to the original number
Factor Relationship Process
15
Factor
3×5
Break
×4
Multiply
3×(5×4)
Regroup
60
Total
Factor Pair Examples
12: (1,12), (2,6), (3,4)
18: (1,18), (2,9), (3,6)
24: (1,24), (2,12), (3,8), (4,6)
Using factor relationships makes multiplication easier!
Master Factor Relationships! 🌟
a × b = c
If a = m × n, then a × b = (m × n) × b = m × (n × b)
Factor Multiplication Property
Key Definitions:

Factor: A number that divides another number evenly

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

Factorization: Breaking a number into its factors

Associative Property: (a×b)×c = a×(b×c)

Prime Number: A number with only two factors (1 and itself)

Composite Number: A number with more than two factors

Finding Factor Pairs Method:
  1. Start low: Begin with 1 and check if it divides evenly
  2. Check systematically: Continue with 2, 3, 4, etc.
  3. Stop at square root: Only check up to √n
  4. List pairs: For each divisor, pair with its quotient
  5. Verify: Check that each pair multiplies to the original number
Tip 1: Always check divisibility up to the square root.
Tip 2: Use factor relationships to make multiplication easier.
Tip 3: Remember the associative property: regroup for easier calculation.
Tip 4: Always verify that your factor pairs multiply correctly.
Common Errors: Missing factor pairs, forgetting to check divisibility up to square root, miscalculating products.
Success Strategy: Practice systematic approach, verify all pairs, use known facts to solve unknown facts.
Factor Properties:

• Every number: Has at least two factors (1 and itself)

• Factor pairs: Come in pairs that multiply to the original number

• Even numbers: Always have 2 as a factor

• Square numbers: Have one factor that appears twice (like 6×6 for 36)

Factor Concepts & Strategies
a × b = c
If a = m × n, then a × b = m × n × b
Factor Multiplication
Factor Categories:

Prime Factors: Factors that are prime numbers

Proper Factors: All factors except the number itself

Improper Factors: The number itself and 1

Multiplication Strategies Using Factors:
  1. Factor decomposition: Break one number into factors
  2. Regrouping: Use associative property to rearrange
  3. Known facts: Use familiar multiplication facts
  4. Verification: Check by using standard algorithm
Strategy 1: Start with small numbers to build understanding.
Strategy 2: Look for factors that result in round numbers.
Strategy 3: Practice with perfect squares and common multiples.
Strategy 4: Use factor trees to visualize factorization.
Key Concepts: Factor relationships make complex multiplication easier.
Real World: Factors apply to arranging objects in rectangles, sharing, and grouping.
Real-World Applications:

• Arranging objects: Organizing items in rectangular arrays

• Sharing: Dividing items equally among groups

• Measurement: Calculating areas using factor relationships

• Planning: Organizing events with equal groupings

Questions & Answers

Question: Why do we need to learn about factor pairs? Isn't it faster to just multiply the numbers directly?

Answer: Learning factor pairs helps you in several ways:

  • Understanding: It shows you why multiplication works
  • Strategy: Sometimes factoring makes multiplication easier (like 15×4 = 3×5×4 = 3×20 = 60)
  • Division: Understanding factors helps with division too
  • Advanced math: Essential for fractions, algebra, and higher math

While direct multiplication works, knowing factors gives you more tools for solving problems. It's like having different keys for different locks!

Factor relationships also help you check if your multiplication is correct.

Question: My child is having trouble finding all the factor pairs. How can I help them be more systematic?

Answer: Here's a systematic approach to help your child:

  • Start at 1: Always begin with 1 and the number itself
  • Go in order: Check 2, 3, 4, 5, etc., systematically
  • Stop early: Only go up to the square root of the number
  • Keep a chart: Write down each pair as you find it
  • Verify: Multiply each pair to confirm it equals the original number

For example, for 24: 1×24, 2×12, 3×8, 4×6. Stop at 4 because 5×5 = 25 > 24.

Practice with smaller numbers first and gradually increase the difficulty.

Question: How can I help my students understand the connection between factors and multiplication strategies?

Answer: The connection is the associative property:

  • Concrete example: 15×4 = (3×5)×4 = 3×(5×4) = 3×20 = 60
  • Visual models: Use arrays to show how factoring changes the arrangement
  • Real-world connection: If you have 15 groups of 4, that's the same as 3 groups of (5×4)
  • Pattern recognition: Show that this works for any factorization

Emphasize that factoring is just applying the associative property. This makes multiplication of larger numbers manageable by using familiar facts. Students who understand this connection become more flexible problem solvers.

The better students understand factors, the more efficient they'll be with multiplication.