Determining Factors and Multiples Answer Key

Complete answer key for determining factors and multiples exercises with detailed solutions and visual infographics.

Answer Key: Determining Factors and Multiples

Key Concepts

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

Multiple: The product of a number and any whole number.

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

Composite Number: A number with more than two factors.

Finding Factors

Method: Divide by numbers 1 to √n.

Rule: If a×b=n, then a and b are factors of n.

Always include: 1 and the number itself.

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

Finding Multiples

Method: Multiply by 1, 2, 3, 4, ...

Rule: Multiples of n are n, 2n, 3n, 4n, ...

Example: First 5 multiples of 6: 6, 12, 18, 24, 30

Infinite: There are infinitely many multiples of any number.

Answer Verification

Factor Check: Multiply factor pairs to verify they equal the original number.

Multiple Check: Divide to verify the result is a whole number.

Complete List: Ensure no factors/multiples are missed.

Order: List factors/multiples in ascending order.

1
Identify Task
2
Apply Method
3
Calculate
4
Verify
5
Record Answer
ANSWER KEY PRINCIPLE: Always verify your answers by checking that factors multiply to the original number and multiples divide evenly by the base number!
Answer Key: Basic Factors and Multiples
1 Finding Factors Easy
ANSWER KEY Example 1
FACTOR PROBLEM Find all factors of 18.
Definition:

Factors: Numbers that divide evenly into another number. For example, the factors of 18 are numbers that divide 18 without leaving a remainder.

Method to find factors:
  1. Start with 1 and the number itself
  2. Check each number up to the square root of the number
  3. If a number divides evenly, add both the divisor and the quotient
  4. List all unique factors
Task
Find factors of 18
Answer
1, 2, 3, 6, 9, 18
Step 1: Find all factors of 18

1×18=18, 2×9=18, 3×6=18

So factors are: 1, 2, 3, 6, 9, 18

Step 2: Verify each factor pair

1×18=18 ✓, 2×9=18 ✓, 3×6=18 ✓

Step 3: Check for completeness

√18 ≈ 4.2, so we only need to check up to 4

4 doesn't divide 18 evenly, so we're done

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

All factors of 18: 1, 2, 3, 6, 9, 18

Verification:

• Factor identification: Find all numbers that divide 18 evenly

• Pair verification: Each pair multiplies to 18

• Completeness check: No other factors exist

🌱
Factor Tree
18
2×9
2×3×3
📊
Factor Pairs
1×18
2×9
3×6
2 Finding Multiples Medium
ANSWER KEY Example 2
MULTIPLE PROBLEM Find the first 6 multiples of 7.
Definition:

Multiples: The product of a number and any whole number. For example, the multiples of 7 are 7×1, 7×2, 7×3, etc.

Task
Find first 6 multiples of 7
Answer
7, 14, 21, 28, 35, 42
Step 1: Multiply 7 by 1 to 6

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

Step 2: Verify each multiple

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

All divisions result in whole numbers

Step 3: Check for correctness

Each multiple is 7 more than the previous: 7, 14, 21, 28, 35, 42

First 6 multiples of 7: 7, 14, 21, 28, 35, 42
Final answer:

The first 6 multiples of 7: 7, 14, 21, 28, 35, 42

Verification:

• Multiplication method: Multiply base number by 1, 2, 3, etc.

• Division check: Each multiple divides evenly by 7

• Pattern check: Each multiple is 7 more than the previous

🔄
Multiples Pattern
7×1
7×2
7×3
7×4
7×5
🔢
Results
7
14
21
28
35
3 Mixed Practice Easy
ANSWER KEY Example 3
MIXED PROBLEM Find all factors of 20 and the first 5 multiples of 4.
Definition:

Factors: Numbers that divide evenly into another number.

Multiples: The product of a number and any whole number.

Task 1
Factors of 20
Answer 1
1, 2, 4, 5, 10, 20
Task 2
First 5 multiples of 4
Answer 2
4, 8, 12, 16, 20
Step 1: Find all factors of 20

1×20=20, 2×10=20, 4×5=20

So factors are: 1, 2, 4, 5, 10, 20

Step 2: Find first 5 multiples of 4

4×1=4, 4×2=8, 4×3=12, 4×4=16, 4×5=20

So multiples are: 4, 8, 12, 16, 20

Step 3: Verify answers

Factors: 1×20=20, 2×10=20, 4×5=20 ✓

Multiples: 4÷4=1, 8÷4=2, 12÷4=3, 16÷4=4, 20÷4=5 ✓

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

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

First 5 multiples of 4: 4, 8, 12, 16, 20

Verification:

• Factor check: Each pair multiplies to 20

• Multiple check: Each divides evenly by 4

• Order: Both lists in ascending order

🌱
Factors of 20
1
2
4
5
10
🔄
Multiples of 4
4
8
12
16
20
Answer Key: Intermediate Factors and Multiples
Factor × Factor = Product
Factors Relationship
Factor Identification
1, 2, 3, 6, 9, 18
Factors of 18
Multiples
7, 14, 21, 28, 35, 42
First 6 multiples of 7
Mixed Results
1,2,4,5,10,20 | 4,8,12,16,20
Factors of 20 | Multiples of 4
Key definitions:

Factors: Numbers that divide evenly into another number

Multiples: Products of a number and any integer

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

Composite Number: A number with more than two factors

Answer key methodology:
  1. Identify task: Determine if finding factors or multiples
  2. Apply method: Use systematic approach for factors or multiplication for multiples
  3. Calculate: Perform necessary mathematical operations
  4. Verify: Check that answers satisfy the original conditions
  5. Record: Present answers in correct format
Tip 1: When finding factors, work systematically from 1 up to the square root.
Tip 2: For multiples, multiply the base number by 1, 2, 3, etc.
Tip 3: Always verify your answers by checking the reverse operation.
Tip 4: List factors and multiples in ascending order for clarity.
Answer Key: Advanced Factors and Multiples
4 Prime Factorization Medium
ANSWER KEY Example 4
PRIME FACTORIZATION Find the prime factorization of 36 and list all its factors.
Definition:

Prime Factorization: Expressing a number as a product of prime numbers.

Task 1
Prime factorization of 36
Answer 1
2² × 3²
Task 2
All factors of 36
Answer 2
1, 2, 3, 4, 6, 9, 12, 18, 36
Step 1: Find prime factorization of 36

36 = 2 × 18 = 2 × 2 × 9 = 2 × 2 × 3 × 3 = 2² × 3²

Step 2: Find all factors using prime factorization

Possible combinations: 2⁰×3⁰=1, 2¹×3⁰=2, 2²×3⁰=4, 2⁰×3¹=3, 2¹×3¹=6, 2²×3¹=12, 2⁰×3²=9, 2¹×3²=18, 2²×3²=36

Step 3: Verify all factors

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

Prime factorization: 2² × 3² | Factors: 1, 2, 3, 4, 6, 9, 12, 18, 36
Final answer:

Prime factorization of 36: 2² × 3²

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

Verification:

• Prime check: All factors are prime numbers or products of primes

• Multiplication check: Each factor pair multiplies to 36

• Completeness: All possible combinations included

⚛️
Prime Factorization
36
2×18
2×2×9
2²×3²
📊
All Factors
1
2
3
4
6
5 Least Common Multiple Hard
ANSWER KEY Example 5
LCM PROBLEM Find the LCM of 12 and 18 using their prime factorizations and list the first 4 multiples of the LCM.
Definition:

Least Common Multiple (LCM): The smallest positive number that is a multiple of two or more numbers.

Task 1
Prime factorization of 12
Answer 1
2² × 3
Task 2
Prime factorization of 18
Answer 2
2 × 3²
Task 3
LCM of 12 and 18
Answer 3
36
Step 1: Find prime factorization of 12

12 = 2 × 6 = 2 × 2 × 3 = 2² × 3

Step 2: Find prime factorization of 18

18 = 2 × 9 = 2 × 3 × 3 = 2 × 3²

Step 3: Find LCM using prime factorizations

Take highest power of each prime: 2² × 3² = 4 × 9 = 36

Step 4: Find first 4 multiples of 36

36×1=36, 36×2=72, 36×3=108, 36×4=144

Step 5: Verify LCM

36 ÷ 12 = 3 (whole number) ✓, 36 ÷ 18 = 2 (whole number) ✓

No smaller number divides both 12 and 18 evenly

LCM of 12 and 18: 36 | First 4 multiples: 36, 72, 108, 144
Final answer:

Prime factorization of 12: 2² × 3

Prime factorization of 18: 2 × 3²

LCM of 12 and 18: 36

First 4 multiples of 36: 36, 72, 108, 144

Verification:

• Prime factorization: Each number expressed as product of primes

• LCM method: Take highest power of each prime factor

• Division check: LCM divides both original numbers evenly

🔄
LCM Process
12 = 2²×3
18 = 2×3²
LCM = 2²×3²
= 36
🔍
Multiples of 36
36×1
36×2
36×3
36×4
Answer Key: Key Concepts and Strategies
LCM(a,b) = (a × b) / GCD(a,b)
LCM Formula
Key definitions:

Factor: A number that divides another number evenly

Multiple: The product of a number and an integer

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

Composite Number: A number with more than two factors

Answer key strategies:
  1. Understand: Read the problem carefully and identify what is being asked
  2. Plan: Decide which mathematical concept to apply (factors or multiples)
  3. Solve: Apply the appropriate method systematically
  4. Check: Verify your answer makes sense in the context of the problem
  5. Record: Present your answer clearly and completely
Tip 1: To find factors, try dividing by small prime numbers first.
Tip 2: For multiples, skip counting helps identify them quickly.
Tip 3: Always verify your answers by checking the reverse operation.
Tip 4: List answers in ascending order for clarity and organization.

Common errors: Forgetting 1 and the number itself as factors, missing multiples, incorrect prime factorization.
Key to success: Practice systematic approaches, master basic multiplication facts, and verify all answers.
Essential formulas and rules:

• Factor pairs: If a×b = n, then a and b are factors of n

• Multiples: Multiples of n are n, 2n, 3n, 4n, ...

• Prime factorization: Express number as product of prime factors

• LCM: Take highest power of each prime factor

• Verification: Always check answers using reverse operations

Answer Key Visualization: Factors and Multiples
ANSWER KEY Example 6: Visualizing Factors and Multiples
Compare the factors and multiples of different numbers:
Number A: 12 (factors: 1, 2, 3, 4, 6, 12) (first 5 multiples: 12, 24, 36, 48, 60)
Number B: 15 (factors: 1, 3, 5, 15) (first 5 multiples: 15, 30, 45, 60, 75)
Number C: 8 (factors: 1, 2, 4, 8) (first 5 multiples: 8, 16, 24, 32, 40)
A:12
B:15
C:8
A:24
B:30
C:16
A:36
B:45
C:24

Analysis: Different numbers have different numbers of factors and different multiple sequences.

  • Number A (12): Has 6 factors, more composite
  • Number B (15): Has 4 factors, semi-prime (product of two primes)
  • Number C (8): Has 4 factors, power of prime (2³)

Answer Key Questions & Answers

Question: I don't understand how to find all the factors of a number. Can you explain the systematic way to do it?

Answer: Great question! Here's the systematic approach to find all factors of a number:

  1. Start with 1 and the number itself (since 1 × number = the number)
  2. Check 2: Does 2 divide the number evenly? If yes, write down 2 and its partner
  3. Continue checking 3, 4, 5, etc., up to the square root of the number
  4. For each divisor found, write down both the divisor and its partner

Example with 18:

  • 1 × 18 = 18 → factors: 1, 18
  • 2 × 9 = 18 → factors: 2, 9
  • 3 × 6 = 18 → factors: 3, 6
  • 4 doesn't divide 18 evenly
  • Since √18 ≈ 4.2, we stop here

All factors of 18: 1, 2, 3, 6, 9, 18

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

Answer: Unlike factors, there are infinitely many multiples of any number. However, you can find as many as needed by multiplying the number by 1, 2, 3, 4, etc.:

Example with multiples of 6:

  • 6×1 = 6
  • 6×2 = 12
  • 6×3 = 18
  • 6×4 = 24
  • 6×5 = 30
  • And so on infinitely...

To verify: Each multiple should divide evenly by 6 (6÷6=1, 12÷6=2, 18÷6=3, etc.)

Question: How do I verify my answers when finding factors and multiples?

Answer: Here are verification methods for both factors and multiples:

For factors:

  • Multiply each factor by its corresponding factor to see if you get the original number
  • Divide the original number by each factor to see if the result is a whole number

For multiples:

  • Divide each multiple by the original number to see if the result is a whole number
  • Check that each multiple is the original number times a whole number

Example: For factors of 20 (1, 2, 4, 5, 10, 20): 1×20=20, 2×10=20, 4×5=20 ✓

Example: For multiples of 5 (5, 10, 15, 20): 5÷5=1, 10÷5=2, 15÷5=3, 20÷5=4 ✓