Free Worksheet: Multiples and Factors for Grade 4

Complete free worksheet with comprehensive practice problems and detailed solutions for mastering multiples and factors.

Worksheet Problems - Basic Level
1 Finding Factors
Problem 1
List all factors of 20. Show your work step by step.
Key Concept:

Factor: A number that divides another number evenly without remainder

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

Solution Method:
  1. Start with 1 and the number itself
  2. Test each number from 2 up to the square root of the number
  3. If a number divides evenly, record both the divisor and quotient
  4. List all factors in ascending order
Number
20
Factor Pairs
1×20, 2×10, 4×5
Step 1: Start with 1 and 20

1 × 20 = 20, so 1 and 20 are factors

Step 2: Test 2

20 ÷ 2 = 10, so 2 and 10 are factors

Step 3: Test 3

20 ÷ 3 = 6.67... (not a whole number), so 3 is not a factor

Step 4: Test 4

20 ÷ 4 = 5, so 4 and 5 are factors

Step 5: Test 5

Since 4×5=20, we already have this pair

Step 6: Stop at √20 ≈ 4.5

We've found all factor pairs

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

The factors of 20 are: 1, 2, 4, 5, 10, 20

Important Notes:

• Division: A factor divides the number evenly

• Pairing: Factors come in pairs that multiply to the original number

• Efficiency: Only test up to the square root of the number

2 Identifying Multiples
Problem 2
Circle the multiples of 7 from this list: 14, 21, 25, 28, 35, 40.
Key Concept:

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

Examples: Multiples of 7 are: 7×1=7, 7×2=14, 7×3=21, etc.

Number
7
List
14, 21, 25, 28, 35, 40
Multiples
14, 21, 28, 35
Step 1: Check 14

14 ÷ 7 = 2 (whole number), so 14 is a multiple of 7

Step 2: Check 21

21 ÷ 7 = 3 (whole number), so 21 is a multiple of 7

Step 3: Check 25

25 ÷ 7 = 3.57... (not a whole number), so 25 is NOT a multiple of 7

Step 4: Check 28

28 ÷ 7 = 4 (whole number), so 28 is a multiple of 7

Step 5: Check 35

35 ÷ 7 = 5 (whole number), so 35 is a multiple of 7

Step 6: Check 40

40 ÷ 7 = 5.71... (not a whole number), so 40 is NOT a multiple of 7

Multiples of 7: 14, 21, 28, 35
Final Answer:

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

Important Notes:

• Division test: If division results in a whole number, it's a multiple

• Multiplication pattern: Multiples follow the pattern: n×1, n×2, n×3...

• Remainder check: If there's a remainder, it's not a multiple

3 Prime vs Composite
Problem 3
Is 19 a prime or composite number? Explain your reasoning.
Key Concepts:

Prime number: A number greater than 1 with exactly two factors: 1 and itself

Composite number: A number greater than 1 with more than two factors

Number
19
Type
Prime
Factors
1, 19
Step 1: Test divisibility by 2

19 ÷ 2 = 9.5 (not a whole number), so 2 is not a factor

Step 2: Test divisibility by 3

19 ÷ 3 = 6.33... (not a whole number), so 3 is not a factor

Step 3: Test divisibility by 4

19 ÷ 4 = 4.75 (not a whole number), so 4 is not a factor

Step 4: Test divisibility by 5

19 ÷ 5 = 3.8 (not a whole number), so 5 is not a factor

Step 5: Stop at √19 ≈ 4.4

No factors found between 2 and 4, so only factors are 1 and 19

Step 6: Classify the number

Since 19 has exactly two factors (1 and 19), it is prime

19 is a prime number because its only factors are 1 and 19
Final Answer:

19 is a prime number because it has exactly two factors: 1 and 19

Important Notes:

• Prime definition: Exactly two factors (1 and itself)

• Composite definition: More than two factors

• Efficiency: Only test up to the square root of the number

Worksheet Problems - Intermediate Level
Practice Makes Perfect
Learning Formula
Problem 4
Common Factors
Find shared factors
Problem 5
Least Common Multiple
Find smallest multiple
Problem 6
Prime Factorization
Break into primes
Intermediate Concepts:

Common Factors: Factors shared by two or more numbers

Least Common Multiple (LCM): The smallest multiple shared by two or more numbers

Prime Factorization: Expressing a number as a product of prime factors

Greatest Common Factor (GCF): The largest factor shared by two or more numbers

Solution Strategies:
  1. Find all factors: For each number individually
  2. Compare lists: Identify common elements
  3. List multiples: Until finding the first common one
  4. Verify results: Check by multiplication or division
Tip 1: Always organize your work in lists to avoid missing factors.
Tip 2: Use divisibility rules to quickly identify potential factors.
Tip 3: For LCM, stop as soon as you find the first common multiple.
Tip 4: Always check your answer by multiplying to verify.
Common errors: Missing factors due to lack of organization, forgetting to check divisibility by 1 and the number itself.
Success strategies: Systematic approach, careful verification, and organized work.
Important Rules:

• Common factors are always less than or equal to the smaller number

• LCM is always greater than or equal to the larger number

• Prime numbers have exactly two factors

• Composite numbers have more than two factors

• The number 1 is a factor of every number

Worksheet Problems - Advanced Level
4 Common Factors
Problem 4
Find all common factors of 30 and 45. What is the greatest common factor?
Key Concept:

Common factors: Factors that two or more numbers share

Greatest Common Factor (GCF): The largest of the common factors

Numbers
30 and 45
Factors of 30
1, 2, 3, 5, 6, 10, 15, 30
Factors of 45
1, 3, 5, 9, 15, 45
Common factors
1, 3, 5, 15
Step 1: Find factors of 30

1×30, 2×15, 3×10, 5×6 → Factors: 1, 2, 3, 5, 6, 10, 15, 30

Step 2: Find factors of 45

1×45, 3×15, 5×9 → Factors: 1, 3, 5, 9, 15, 45

Step 3: Compare the lists

Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30

Factors of 45: 1, 3, 5, 9, 15, 45

Common factors: 1, 3, 5, 15

Step 4: Identify the GCF

The greatest common factor is 15

Common factors: 1, 3, 5, 15 | GCF: 15
Final Answer:

The common factors of 30 and 45 are: 1, 3, 5, 15

The Greatest Common Factor (GCF) is 15

Important Notes:

• Systematic approach: Find factors of each number separately

• Comparison: Look for factors that appear in both lists

• Organization: List factors in ascending order to avoid missing any

5 Least Common Multiple
Problem 5
Find the LCM of 9 and 12.
Key Concept:

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

Numbers
9 and 12
Multiples of 9
9, 18, 27, 36, 45...
Multiples of 12
12, 24, 36, 48, 60...
LCM
36
Step 1: List multiples of 9

9×1=9, 9×2=18, 9×3=27, 9×4=36, 9×5=45, 9×6=54

Step 2: List multiples of 12

12×1=12, 12×2=24, 12×3=36, 12×4=48

Step 3: Find the first common multiple

Multiples of 9: 9, 18, 27, 36, 45, 54...

Multiples of 12: 12, 24, 36, 48...

The first common multiple is 36

Step 4: Verify the answer

36 ÷ 9 = 4 and 36 ÷ 12 = 3, so 36 is divisible by both 9 and 12

LCM of 9 and 12 is 36
Final Answer:

The LCM of 9 and 12 is 36

Important Notes:

• List method: Write out multiples until you find a match

• Verification: Check that the LCM is divisible by both numbers

• Efficiency: Stop as soon as you find the first common multiple

Worksheet Review and Key Takeaways
Factor × Factor = Product
Factor-Multiple Relationship
Core Concepts Summary:

Factors: Numbers that divide another number evenly without remainder. Factors are usually smaller than the number being factored. Example: Factors of 12 are 1, 2, 3, 4, 6, 12 because 1×12=12, 2×6=12, 3×4=12.

Multiples: Numbers that result from multiplying a given number by any whole number. Multiples are usually larger than the original number. Example: Multiples of 5 are 5, 10, 15, 20, 25... because 5×1=5, 5×2=10, 5×3=15, etc.

Prime Numbers: Numbers greater than 1 that have exactly two factors: 1 and themselves. Examples: 2, 3, 5, 7, 11, 13, 17, 19, 23... The number 2 is the only even prime number.

Composite Numbers: Numbers greater than 1 that have more than two factors. Examples: 4, 6, 8, 9, 10, 12, 14, 15, 16... The number 4 is the first composite number.

Relationship: If a is a factor of b, then b is a multiple of a. For example, 3 is a factor of 12, and 12 is a multiple of 3. This relationship works both ways.

Systematic Finding Methods:
  1. Finding Factors: Test divisibility from 1 up to the square root of the number. For each divisor found, include both the divisor and quotient.
  2. Finding Multiples: Multiply the number by 1, 2, 3, 4, etc. Multiples continue infinitely.
  3. Prime Testing: Check divisibility up to the square root, looking for factors other than 1 and itself.
  4. Common Factors: Find factors of each number and identify overlaps in the lists.
  5. LCM Finding: List multiples until finding the first common one, or use prime factorization.
  6. Verification: Always check results by multiplying or dividing to ensure accuracy.
Tip 1: Factors come in pairs that multiply to the original number (e.g., 3×4=12, so 3 and 4 are factor pairs of 12).
Tip 2: To find multiples quickly, count by the number (skip counting): 6, 12, 18, 24, 30...
Tip 3: The GCF is always less than or equal to the smaller number, and the LCM is always greater than or equal to the larger number.
Tip 4: Remember: Factors go INTO the number, multiples come OUT of the number.
Tip 5: Use divisibility rules to quickly identify potential factors (by 2, 3, 5, etc.).
Tip 6: Every number is both a factor and multiple of itself.
Common errors: Forgetting 1 and the number itself as factors, confusing factors with multiples, misidentifying primes.
Key insight: Understanding factors and multiples builds foundational skills for fraction operations and algebra.
Memory aid: Factor trees help visualize how numbers break down into their prime components.
Pattern: Prime numbers have exactly two factors, composite numbers have more than two.
Essential Rules and Properties:

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

• The number 1 is a factor of every number

• Prime numbers have exactly two factors: 1 and themselves

• Composite numbers have more than two factors

• The number 1 is neither prime nor composite

• Divisibility by 2: number ends in even digit (0, 2, 4, 6, 8)

• Divisibility by 3: sum of digits is divisible by 3

• Divisibility by 5: number ends in 0 or 5

• Divisibility by 10: number ends in 0

• Factors of a number are finite in quantity

• Multiples of a number are infinite in quantity

Worksheet Performance Tracker
Exercise 6: Skill Assessment
Track your progress as you complete the worksheet exercises.

Progress Tracking: Monitor your improvement in factors and multiples skills.

  • Factors identification accuracy increases with practice
  • Multiples recognition becomes faster
  • Prime/composite classification improves
  • Common factors and LCM calculations become more efficient
  • Word problems become more manageable

Questions & Answers

Question: This worksheet seems hard. How can I make it easier to understand?

Answer: Here are strategies to make the worksheet easier:

  • Start with easier problems: Begin with basic factor identification
  • Use manipulatives: Counters or blocks to visualize factors
  • Work step-by-step: Don't rush through the problems
  • Check your work: Verify each answer before moving on
  • Ask for help: Don't hesitate to ask a teacher or parent
  • Take breaks: Rest if you feel frustrated

Remember that practice makes progress. Each problem you solve helps build your understanding for the next one.

Try to relate the problems to real-life situations like sharing items or arranging objects!

Question: How can I help my child with this free worksheet at home?

Answer: Here's how to support your child with the worksheet:

  • Review concepts: Go over definitions before starting
  • Work together: Complete the first few problems together
  • Use visual aids: Draw arrays or use objects to represent factors
  • Encourage thinking: Ask questions rather than giving answers
  • Check progress: Review completed problems together
  • Provide encouragement: Celebrate effort and improvement

Create a quiet workspace and allow your child to work at their own pace. Focus on understanding rather than speed.

Remember that making mistakes is part of learning - use them as opportunities to understand concepts better!

Question: What if I get stuck on a problem in the worksheet? Should I skip it?

Answer: Here's what to do when you're stuck:

  • Read carefully: Make sure you understand what the problem is asking
  • Try a simpler version: Use smaller numbers to understand the concept
  • Look back: Check the examples provided in the worksheet
  • Take a break: Step away for a few minutes and return with fresh eyes
  • Ask for help: Talk to a teacher, parent, or friend
  • Move on temporarily: Skip and come back later if needed

Don't skip problems entirely - try to attempt them even if you're unsure. Sometimes partial work leads to insights.

Remember that struggling with a problem is normal and part of the learning process!