Solved Exercises on Factors and Multiples in Grade 6

Master factors and multiples: prime factorization, GCF, LCM, and divisibility rules through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Finding Factors
Exercise 1
Find all factors of 48. List them in order from least to greatest. How many factors does 48 have?
Definition:

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

Factor finding method:
  1. Start with 1: 1 is always a factor
  2. Check divisibility: Test each number up to the square root
  3. Find factor pairs: For each factor found, find its partner
  4. List in order: Organize factors from smallest to largest
Finding Factors of 48:

1 × 48 = 48

2 × 24 = 48

3 × 16 = 48

4 × 12 = 48

6 × 8 = 48

No more pairs (7 and 8 are consecutive, so we stop)

Step 1: Start with 1 and the number itself

1 and 48 are factors

Step 2: Test divisibility by 2

48 ÷ 2 = 24, so 2 and 24 are factors

Step 3: Test divisibility by 3

48 ÷ 3 = 16, so 3 and 16 are factors

Step 4: Test divisibility by 4

48 ÷ 4 = 12, so 4 and 12 are factors

Step 5: Test divisibility by 5

48 ÷ 5 = 9.6, so 5 is not a factor

Step 6: Test divisibility by 6

48 ÷ 6 = 8, so 6 and 8 are factors

Step 7: Stop at √48 ≈ 6.9

Since 7² = 49 > 48, we don't need to test beyond 6

Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48
Final answer:

The factors of 48 are: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48. There are 10 factors.

Applied rules:

• Factor pairs: If a is a factor of n, then n/a is also a factor

• Square root limit: Only check up to √n to find all factors

• Divisibility test: A number divides evenly if there's no remainder

2 Finding Multiples
Exercise 2
List the first six multiples of 7. Then find the least common multiple (LCM) of 7 and 12.
Definition:

Multiple: The product of a number and any whole number

Multiples of 7:

7 × 1 = 7

7 × 2 = 14

7 × 3 = 21

7 × 4 = 28

7 × 5 = 35

7 × 6 = 42

Finding LCM of 7 and 12:

Method 1: List multiples until common one appears

7: 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, ...

12: 12, 24, 36, 48, 60, 72, 84, ...

First common multiple: 84

Step 1: Find first six multiples of 7

7 × 1 = 7

7 × 2 = 14

7 × 3 = 21

7 × 4 = 28

7 × 5 = 35

7 × 6 = 42

Step 2: Find multiples of 12

12 × 1 = 12

12 × 2 = 24

12 × 3 = 36

12 × 4 = 48

12 × 5 = 60

12 × 6 = 72

12 × 7 = 84

Step 3: Identify the least common multiple

Compare lists: 84 is the first number that appears in both lists

Step 4: Verify the result

84 ÷ 7 = 12 ✓

84 ÷ 12 = 7 ✓

Multiples of 7: 7, 14, 21, 28, 35, 42; LCM(7,12) = 84
Final answer:

The first six multiples of 7 are: 7, 14, 21, 28, 35, 42. The LCM of 7 and 12 is 84.

Applied rules:

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

• LCM: Smallest number that is a multiple of both numbers

• Verification: LCM should be divisible by both original numbers

3 Prime Factorization
Exercise 3
Find the prime factorization of 60. Express your answer using exponential notation. Then use it to find the GCF of 60 and 45.
Definition:

Prime factorization: Expressing a number as a product of prime numbers

Prime Factorization of 60:

60 = 2 × 30

30 = 2 × 15

15 = 3 × 5

So: 60 = 2 × 2 × 3 × 5 = 2² × 3¹ × 5¹

Finding GCF of 60 and 45:

60 = 2² × 3¹ × 5¹

45 = 3² × 5¹

GCF = Take lowest power of common primes: 3¹ × 5¹ = 15

Step 1: Begin with 60 and find first factor pair

60 = 2 × 30 (2 is prime, 30 is composite)

Step 2: Factor the composite number

30 = 2 × 15 (2 is prime, 15 is composite)

Step 3: Continue factoring composite numbers

15 = 3 × 5 (both 3 and 5 are prime)

Step 4: Write in exponential notation

60 = 2 × 2 × 3 × 5 = 2² × 3¹ × 5¹

Step 5: Find prime factorization of 45

45 = 3 × 15 = 3 × 3 × 5 = 3² × 5¹

Step 6: Find GCF using prime factorizations

60 = 2² × 3¹ × 5¹

45 = 3² × 5¹

Common primes: 3 and 5

Lowest powers: 3¹ and 5¹

GCF = 3¹ × 5¹ = 15

Prime factorization: 2² × 3 × 5; GCF(60,45) = 15
Final answer:

The prime factorization of 60 is 2² × 3 × 5. The GCF of 60 and 45 is 15.

Applied rules:

• Prime factorization: Divide by smallest prime factors first

• Exponential notation: Count repeated factors using exponents

• GCF method: Take lowest power of common prime factors

Factors and Multiples Properties
GCF(a,b) × LCM(a,b) = a × b
Fundamental Relationship
Concept 1
Factors
Numbers that divide evenly into another
Concept 2
Multiples
Products of a number and whole numbers
Concept 3
Prime
Only divisible by 1 and itself
Key definitions:

Factor: A number that divides evenly into another number (e.g., 3 is a factor of 12)

Multiple: The product of a number and any whole number (e.g., 12 is a multiple of 3)

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

Composite number: A number with more than two factors

Prime factorization: Expressing a number as a product of prime numbers

Factor Finding Process:
  1. Start systematically: Begin with 1 and check consecutive numbers
  2. Use divisibility rules: Apply shortcuts to identify factors quickly
  3. Find pairs: For each factor found, identify its complementary factor
  4. Stop at square root: Only check up to √n to find all factors
  5. Organize: List factors in ascending order
Tip 1: Use divisibility rules to quickly identify factors (2: even, 3: sum of digits divisible by 3, etc.).
Tip 2: When finding GCF, list all factors of both numbers and find the largest common one.
Tip 3: For LCM, use prime factorization method for larger numbers.
Tip 4: Remember: 1 is a factor of every number, and every number is a factor of itself.
Common errors: Confusing factors and multiples, forgetting to include 1 and the number itself, stopping before finding all factors.
Success habits: Using systematic approaches, double-checking work, practicing divisibility rules.
Divisibility Rules:

• Divisible by 2: Last digit is even (0, 2, 4, 6, 8)

• Divisible by 3: Sum of digits is divisible by 3

• Divisible by 4: Last two digits form a number divisible by 4

• Divisible by 5: Last digit is 0 or 5

• Divisible by 6: Divisible by both 2 and 3

• Divisible by 9: Sum of digits is divisible by 9

• Divisible by 10: Last digit is 0

Solution: Exercises 4 to 5
4 Greatest Common Factor
Exercise 4
Find the GCF of 72 and 108 using both the listing method and prime factorization method. Which method is more efficient for these numbers?
Definition:

Greatest Common Factor (GCF): The largest number that divides evenly into two or more numbers

Listing Method:

Factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72

Factors of 108: 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108

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

GCF = 36

Prime Factorization Method:

72 = 2³ × 3²

108 = 2² × 3³

Common primes: 2 and 3

Lowest powers: 2² and 3²

GCF = 2² × 3² = 4 × 9 = 36

Step 1: Find factors of 72

72 = 1×72, 2×36, 3×24, 4×18, 6×12, 8×9

Factors: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72

Step 2: Find factors of 108

108 = 1×108, 2×54, 3×36, 4×27, 6×18, 9×12

Factors: 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108

Step 3: Identify common factors

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

Step 4: Prime factorization of 72

72 ÷ 2 = 36

36 ÷ 2 = 18

18 ÷ 2 = 9

9 ÷ 3 = 3

3 ÷ 3 = 1

So 72 = 2³ × 3²

Step 5: Prime factorization of 108

108 ÷ 2 = 54

54 ÷ 2 = 27

27 ÷ 3 = 9

9 ÷ 3 = 3

3 ÷ 3 = 1

So 108 = 2² × 3³

Step 6: Find GCF using prime factorization

Take lowest power of common primes: 2² × 3² = 36

GCF of 72 and 108: 36
Final answer:

The GCF of 72 and 108 is 36. For these numbers, the prime factorization method is more efficient as listing all factors takes longer.

Applied rules:

• Listing method: Systematically find all factors of each number

• Prime factorization method: Take lowest power of common prime factors

• Efficiency: Prime factorization is often faster for larger numbers

5 Least Common Multiple Word Problem
Exercise 5
Two buses leave the station at the same time. Bus A returns every 15 minutes, and Bus B returns every 20 minutes. After how many minutes will both buses return to the station at the same time again?
Definition:

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

Finding LCM of 15 and 20:

Method 1: List multiples

15: 15, 30, 45, 60, 75, ...

20: 20, 40, 60, 80, ...

First common multiple: 60

Method 2: Prime factorization

15 = 3 × 5

20 = 2² × 5

LCM = 2² × 3 × 5 = 60

Step 1: Understand the problem

Bus A returns every 15 minutes

Bus B returns every 20 minutes

Find when both will return simultaneously

Step 2: Recognize this is an LCM problem

We need the smallest number that is a multiple of both 15 and 20

Step 3: Find prime factorization of each number

15 = 3 × 5

20 = 2² × 5

Step 4: Find LCM using prime factorization

Take highest power of each prime: 2² × 3¹ × 5¹ = 4 × 3 × 5 = 60

Step 5: Verify the answer

60 ÷ 15 = 4 (Bus A returns 4 times)

60 ÷ 20 = 3 (Bus B returns 3 times)

Both buses return after 60 minutes

Both buses return after 60 minutes
Final answer:

Both buses will return to the station at the same time after 60 minutes.

Applied rules:

• Word problem recognition: Simultaneous events → LCM

• LCM definition: Smallest common multiple

• Verification: Check that the result is divisible by both original numbers

Complete Summary: Factors and Multiples
a × b = GCF(a,b) × LCM(a,b)
Relationship Formula
Key definitions:

Factor: A number that divides evenly into another number 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

Greatest Common Factor (GCF): The largest number that divides evenly into two or more numbers

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

Complete methodology:
  1. Factor identification: Systematically find all factors using divisibility rules
  2. Prime factorization: Break numbers down into products of primes
  3. GCF calculation: Use either listing method or prime factorization method
  4. LCM calculation: Use either listing method or prime factorization method
  5. Verification: Check that your answers satisfy the original problem conditions
Tip 1: For large numbers, use prime factorization method rather than listing all factors.
Tip 2: Remember that 1 is always a factor and never a multiple (except for 1 itself).
Tip 3: Use the relationship: GCF(a,b) × LCM(a,b) = a × b to verify your answers.
Tip 4: Practice divisibility rules regularly to quickly identify factors.

Important relationships: GCF is never larger than the smaller number; LCM is never smaller than the larger number.
Applications: Simplifying fractions, adding/subtracting fractions, scheduling problems, and factoring polynomials.
Essential Rules:

• Factor finding: Only check up to √n to find all factors

• Prime factorization: Divide by smallest prime factors first

• GCF method: Take lowest power of common prime factors

• LCM method: Take highest power of all prime factors

• Relationship: Product of two numbers = GCF × LCM

Questions & Answers

Question: I sometimes confuse factors and multiples. How can I remember which is which?

Answer: Here are some memory aids:

  • Factors: "Fits in" - factors fit into the number (divide evenly)
  • Multiples: "More" - multiples are bigger (products)
  • Factor example: 3 is a factor of 12 because 3 fits into 12 evenly (12 ÷ 3 = 4)
  • Multiple example: 12 is a multiple of 3 because 3 × 4 = 12 (the result is bigger)

Think of factors as "ingredients" that make up the number, and multiples as "results" of multiplying the number by other numbers.

Another way: Factors are fewer and smaller than the original number (except the number itself), while multiples are infinite and generally larger than the original number.

Practice with concrete examples: For the number 8, factors are 1, 2, 4, 8 (all ≤ 8), and multiples are 8, 16, 24, 32... (all ≥ 8).

Question: When should I use the prime factorization method versus the listing method for finding GCF and LCM?

Answer: Choose your method based on the numbers:

Use listing method when:

  • Numbers are small (less than 20)
  • You can easily list all factors/multiples
  • Numbers have few factors

Use prime factorization method when:

  • Numbers are large
  • Numbers have many factors
  • You need to find GCF/LCM of more than two numbers
  • You want to see the mathematical relationship clearly

Example: For GCF(12, 18), listing is quick: factors of 12: {1,2,3,4,6,12}, factors of 18: {1,2,3,6,9,18}, GCF = 6.

For GCF(48, 72), prime factorization is more efficient: 48 = 2⁴ × 3¹, 72 = 2³ × 3², GCF = 2³ × 3¹ = 24.

Both methods give the same answer - choose the one that's faster for your specific numbers!

Question: How do I know if a number is prime? Do I have to test every number below it?

Answer: You don't need to test every number! Here's the efficient approach:

  1. Test only prime numbers: Check divisibility by 2, 3, 5, 7, 11, 13, etc.
  2. Stop at the square root: Only test up to √n
  3. Use divisibility rules: Apply shortcuts for 2, 3, 5, etc.

Example: To check if 37 is prime:

  • Is 37 even? No, so not divisible by 2
  • Sum of digits: 3 + 7 = 10, not divisible by 3
  • Doesn't end in 0 or 5, so not divisible by 5
  • √37 ≈ 6.08, so only check up to 6
  • 37 ÷ 7 = 5.29... (not whole), so not divisible by 7
  • Since 7² = 49 > 37, we can stop

Therefore, 37 is prime because it has no factors other than 1 and itself.

Remember: 2 is the only even prime number, and 1 is neither prime nor composite.