Comprehensive Practice Problems: Multiples and Factors for Grade 4

Extensive practice problems with detailed solutions and step-by-step guidance for mastering multiples and factors.

Practice Problems - Basic Level
1 Finding Factors
Problem 1
List all factors of 18. 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
18
Factor Pairs
1×18, 2×9, 3×6
Step 1: Start with 1 and 18

1 × 18 = 18, so 1 and 18 are factors

Step 2: Test 2

18 ÷ 2 = 9, so 2 and 9 are factors

Step 3: Test 3

18 ÷ 3 = 6, so 3 and 6 are factors

Step 4: Test 4

18 ÷ 4 = 4.5 (not a whole number), so 4 is not a factor

Step 5: Test 5

18 ÷ 5 = 3.6 (not a whole number), so 5 is not a factor

Step 6: Stop at √18 ≈ 4.2

We've found all factor pairs

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

The factors of 18 are: 1, 2, 3, 6, 9, 18

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
Which numbers from this list are multiples of 6: 12, 18, 24, 30, 35, 36?
Key Concept:

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

Examples: Multiples of 6 are: 6×1=6, 6×2=12, 6×3=18, etc.

Number
6
List
12, 18, 24, 30, 35, 36
Multiples
12, 18, 24, 30, 36
Step 1: Check 12

12 ÷ 6 = 2 (whole number), so 12 is a multiple of 6

Step 2: Check 18

18 ÷ 6 = 3 (whole number), so 18 is a multiple of 6

Step 3: Check 24

24 ÷ 6 = 4 (whole number), so 24 is a multiple of 6

Step 4: Check 30

30 ÷ 6 = 5 (whole number), so 30 is a multiple of 6

Step 5: Check 35

35 ÷ 6 = 5.83... (not a whole number), so 35 is NOT a multiple of 6

Step 6: Check 36

36 ÷ 6 = 6 (whole number), so 36 is a multiple of 6

Multiples of 6: 12, 18, 24, 30, 36
Final Answer:

The multiples of 6 from the list are: 12, 18, 24, 30, 36

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 23 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
23
Type
Prime
Factors
1, 23
Step 1: Test divisibility by 2

23 ÷ 2 = 11.5 (not a whole number), so 2 is not a factor

Step 2: Test divisibility by 3

23 ÷ 3 = 7.67... (not a whole number), so 3 is not a factor

Step 3: Test divisibility by 4

23 ÷ 4 = 5.75 (not a whole number), so 4 is not a factor

Step 4: Test divisibility by 5

23 ÷ 5 = 4.6 (not a whole number), so 5 is not a factor

Step 5: Stop at √23 ≈ 4.8

No factors found between 2 and 4, so only factors are 1 and 23

Step 6: Classify the number

Since 23 has exactly two factors (1 and 23), it is prime

23 is a prime number because its only factors are 1 and 23
Final Answer:

23 is a prime number because it has exactly two factors: 1 and 23

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

Practice Problems - Intermediate Level
Practice × Repetition = Mastery
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

Practice Problems - Advanced Level
4 Common Factors
Problem 4
Find all common factors of 24 and 36. 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
24 and 36
Factors of 24
1, 2, 3, 4, 6, 8, 12, 24
Factors of 36
1, 2, 3, 4, 6, 9, 12, 18, 36
Common factors
1, 2, 3, 4, 6, 12
Step 1: Find factors of 24

1×24, 2×12, 3×8, 4×6 → Factors: 1, 2, 3, 4, 6, 8, 12, 24

Step 2: Find factors of 36

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

Step 3: Compare the lists

Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24

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

Common factors: 1, 2, 3, 4, 6, 12

Step 4: Identify the GCF

The greatest common factor is 12

Common factors: 1, 2, 3, 4, 6, 12 | GCF: 12
Final Answer:

The common factors of 24 and 36 are: 1, 2, 3, 4, 6, 12

The Greatest Common Factor (GCF) is 12

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 8 and 12.
Key Concept:

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

Numbers
8 and 12
Multiples of 8
8, 16, 24, 32, 40...
Multiples of 12
12, 24, 36, 48, 60...
LCM
24
Step 1: List multiples of 8

8×1=8, 8×2=16, 8×3=24, 8×4=32, 8×5=40, 8×6=48

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 8: 8, 16, 24, 32, 40, 48...

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

The first common multiple is 24

Step 4: Verify the answer

24 ÷ 8 = 3 and 24 ÷ 12 = 2, so 24 is divisible by both 8 and 12

LCM of 8 and 12 is 24
Final Answer:

The LCM of 8 and 12 is 24

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

Comprehensive 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

Practice Performance Analysis
Exercise 6: Skill Development
Track your progress as you practice factors and multiples concepts.

Progress Tracking: How your understanding improves with practice over time.

  • 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: I'm having trouble remembering all the factors of numbers. Do you have any tips?

Answer: Here are some helpful strategies for remembering factors:

  • Start with 1 and the number itself: These are always factors
  • Use factor pairs: Remember that factors come in pairs that multiply to the number
  • Test systematically: Go in order from 2 up to the square root of the number
  • Use divisibility rules: Check if the number is divisible by 2, 3, 5, etc.
  • Organize in lists: Write factors in ascending order to avoid missing any
  • Practice regularly: The more you practice, the more automatic it becomes

For example, with 24: start with 1×24, then 2×12, then 3×8, then 4×6. When you reach 5, notice that 24÷5 isn't a whole number, so stop there.

Create factor rainbows or use factor trees to visualize the relationships between factors!

Question: How much practice should my child do each day to master factors and multiples?

Answer: Consistency is more important than duration. Here's a recommended approach:

  • Daily practice: 15-20 minutes per day is ideal
  • Short sessions: Keep attention focused and prevent fatigue
  • Varied problems: Mix different types of factor and multiple problems
  • Review regularly: Revisit concepts weekly to maintain retention
  • Gradual increase: Start with basic problems and progress to advanced ones
  • Quality over quantity: Focus on understanding rather than rushing through problems

Practice should be consistent but not overwhelming. Watch for signs of frustration or fatigue, and adjust accordingly. The goal is to build confidence and understanding.

Remember, it's better to do fewer problems correctly than many problems incorrectly!

Question: Why do I need to learn about factors and multiples? When will I use them?

Answer: Factors and multiples are used in many real-life situations:

  • Cooking: Doubling or halving recipes uses multiples
  • Shopping: Buying items in packages (6-pack, 12-pack) involves factors
  • Time: Hours and minutes work in multiples of 60
  • Money: Counting coins and bills involves factors
  • Arranging things: Organizing books, toys, or chairs
  • Music: Rhythm and beats follow multiple patterns

In math, factors and multiples help with fractions, ratios, percentages, and algebra. They're also important for understanding number patterns and relationships.

These concepts also build logical thinking and problem-solving skills that are valuable in many subjects!