Solved Exercises on Greatest Common Factor (GCF) in Grade 6

Master GCF: listing method, prime factorization, and applications through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Listing Method
Exercise 1
Find the GCF of 24 and 36 by listing all factors of each number and identifying the greatest common factor.
Definition:

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

Listing method:
  1. Find factors of first number: List all numbers that divide it evenly
  2. Find factors of second number: List all numbers that divide it evenly
  3. Identify common factors: Find factors that appear in both lists
  4. Select the greatest: Choose the largest common factor
Factors of 24:
24 = 1 × 24
24 = 2 × 12
24 = 3 × 8
24 = 4 × 6
Factors of 24: {1, 2, 3, 4, 6, 8, 12, 24}
Factors of 36:
36 = 1 × 36
36 = 2 × 18
36 = 3 × 12
36 = 4 × 9
36 = 6 × 6
Factors of 36: {1, 2, 3, 4, 6, 9, 12, 18, 36}
Step 1: Find all factors of 24

Check divisibility: 1, 2, 3, 4, 6, 8, 12, 24

Step 2: Find all factors of 36

Check divisibility: 1, 2, 3, 4, 6, 9, 12, 18, 36

Step 3: Identify common factors

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

Step 4: Find the greatest common factor

The largest number in {1, 2, 3, 4, 6, 12} is 12

Step 5: Verify the result

24 ÷ 12 = 2 ✓

36 ÷ 12 = 3 ✓

GCF(24,36) = 12
Final answer:

The GCF of 24 and 36 is 12.

Applied rules:

• Factor identification: Find all numbers that divide evenly

• Common factor principle: GCF must divide all original numbers

• Verification: Check that GCF divides both numbers evenly

2 Prime Factorization Method
Exercise 2
Find the GCF of 48 and 60 using prime factorization. Show your work step by step.
Definition:

Prime factorization method: Finding GCF by taking the lowest power of common prime factors

Prime Factorization of 48:

48 = 2⁴ × 3¹

48 ÷ 2 = 24

24 ÷ 2 = 12

12 ÷ 2 = 6

6 ÷ 2 = 3

3 ÷ 3 = 1

Prime Factorization of 60:

60 = 2² × 3¹ × 5¹

60 ÷ 2 = 30

30 ÷ 2 = 15

15 ÷ 3 = 5

5 ÷ 5 = 1

Step 1: Find prime factorization of 48

48 = 2⁴ × 3¹

Step 2: Find prime factorization of 60

60 = 2² × 3¹ × 5¹

Step 3: Identify common prime factors

Common primes: 2 and 3

Step 4: Take lowest power of each common prime

For 2: min(4, 2) = 2

For 3: min(1, 1) = 1

Step 5: Calculate GCF

GCF = 2² × 3¹ = 4 × 3 = 12

Step 6: Verify the result

48 ÷ 12 = 4 ✓

60 ÷ 12 = 5 ✓

GCF(48,60) = 12
Final answer:

The GCF of 48 and 60 is 12.

Applied rules:

• Prime factorization: Express each number as product of primes

• Common prime principle: GCF uses only primes common to all numbers

• Lowest power rule: Take minimum exponent for each common prime

3 Three Numbers
Exercise 3
Find the GCF of 24, 36, and 48 using the prime factorization method. Explain your approach.
Definition:

GCF of multiple numbers: The largest number that divides all given numbers evenly

Prime Factorizations:

24 = 2³ × 3¹

36 = 2² × 3²

48 = 2⁴ × 3¹

Step 1: Find prime factorization of 24

24 = 2³ × 3¹

Step 2: Find prime factorization of 36

36 = 2² × 3²

Step 3: Find prime factorization of 48

48 = 2⁴ × 3¹

Step 4: Identify common prime factors in all three

All three have: 2 and 3

Step 5: Take lowest power of each common prime

For 2: min(3, 2, 4) = 2

For 3: min(1, 2, 1) = 1

Step 6: Calculate GCF

GCF = 2² × 3¹ = 4 × 3 = 12

Step 7: Verify the result

24 ÷ 12 = 2 ✓

36 ÷ 12 = 3 ✓

48 ÷ 12 = 4 ✓

GCF(24,36,48) = 12
Final answer:

The GCF of 24, 36, and 48 is 12.

Applied rules:

• Extension principle: Same method applies to multiple numbers

• Universal commonality: Must be common to ALL numbers

• Minimum exponent: Take smallest power among all numbers

GCF Methods and Properties
GCF(a,b) × LCM(a,b) = a × b
Fundamental Relationship
Method 1
Listing
Find all factors
Method 2
Prime Factorization
Lowest power of common primes
Method 3
Euclidean Algorithm
Division-based method
Key definitions:

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

Factor: A number that divides another number evenly without remainder

Common factor: A factor that two or more numbers share

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

GCF Finding Process:
  1. Choose method: Decide between listing or prime factorization
  2. Find factors: Either list all or find prime factorization
  3. Identify commonality: Find factors shared by all numbers
  4. Select maximum: Choose the largest common factor
  5. Verify: Ensure the GCF divides all original numbers
Tip 1: For small numbers (≤ 20), listing method is often faster.
Tip 2: For large numbers, prime factorization method is more efficient.
Tip 3: Always check your answer by dividing each number by the GCF.
Tip 4: The GCF is never larger than the smallest of the original numbers.
Common errors: Forgetting to include 1 as a factor, not finding all factors, taking highest instead of lowest powers.
Success habits: Working systematically, double-checking work, using divisibility rules.
Important Properties:

• GCF(a,b) ≤ min(a,b): GCF is never larger than the smaller number

• GCF(a,a) = a: Number with itself has GCF equal to itself

• GCF(a,1) = 1: Any number with 1 has GCF of 1

• GCF relationship: GCF(a,b) × LCM(a,b) = a × b

• Coprime numbers: If GCF(a,b) = 1, numbers are coprime

Solution: Exercises 4 to 5
4 Real-World Application
Exercise 4
A teacher wants to arrange 48 chairs and 64 desks into equal rows with the same number of items in each row. What is the greatest number of items that can be placed in each row? How many rows will there be?
Definition:

Real-world application: Using GCF to find the largest equal groupings

Finding GCF of 48 and 64:

Prime factorization method:

48 = 2⁴ × 3¹

64 = 2⁶

Common prime: 2

Lowest power: 2⁴ = 16

Step 1: Identify the problem type

We need equal rows for both chairs and desks

This means we need the GCF of 48 and 64

Step 2: Find prime factorization of 48

48 = 2⁴ × 3¹

Step 3: Find prime factorization of 64

64 = 2⁶

Step 4: Identify common prime factors

Only common prime: 2

Step 5: Take lowest power of common primes

Min power of 2: min(4, 6) = 4

GCF = 2⁴ = 16

Step 6: Calculate number of rows

Rows of chairs: 48 ÷ 16 = 3 rows

Rows of desks: 64 ÷ 16 = 4 rows

Total rows: 3 + 4 = 7 rows

16 items per row, 7 rows total
Final answer:

The greatest number of items that can be placed in each row is 16. There will be 3 rows of chairs and 4 rows of desks, for a total of 7 rows.

Applied rules:

• Equal grouping: Real-world problems often require equal divisions

• Maximum efficiency: GCF gives the largest possible equal groupings

• Verification: Check that GCF divides both numbers evenly

5 Advanced Problem
Exercise 5
Find the GCF of 120, 144, and 180. Then use this to determine the largest possible size of square tiles that can be used to tile rectangles of these dimensions without cutting any tiles.
Definition:

Advanced application: Using GCF for optimal tiling solutions

Prime Factorizations:

120 = 2³ × 3¹ × 5¹

144 = 2⁴ × 3²

180 = 2² × 3² × 5¹

Common primes: 2 and 3

Lowest powers: 2² and 3¹

GCF = 2² × 3¹ = 4 × 3 = 12

Step 1: Find prime factorization of 120

120 ÷ 2 = 60

60 ÷ 2 = 30

30 ÷ 2 = 15

15 ÷ 3 = 5

5 ÷ 5 = 1

So 120 = 2³ × 3¹ × 5¹

Step 2: Find prime factorization of 144

144 ÷ 2 = 72

72 ÷ 2 = 36

36 ÷ 2 = 18

18 ÷ 2 = 9

9 ÷ 3 = 3

3 ÷ 3 = 1

So 144 = 2⁴ × 3²

Step 3: Find prime factorization of 180

180 ÷ 2 = 90

90 ÷ 2 = 45

45 ÷ 3 = 15

15 ÷ 3 = 5

5 ÷ 5 = 1

So 180 = 2² × 3² × 5¹

Step 4: Identify common prime factors

Common primes in all three: 2 and 3

Step 5: Take lowest power of each common prime

For 2: min(3, 4, 2) = 2

For 3: min(1, 2, 2) = 1

Step 6: Calculate GCF

GCF = 2² × 3¹ = 4 × 3 = 12

Step 7: Interpret the result for tiling

Largest square tile that fits all rectangles: 12 × 12 inches

This tile will fit evenly into all three dimensions

GCF(120,144,180) = 12
Final answer:

The GCF of 120, 144, and 180 is 12. The largest possible square tile that can be used for tiling is 12 × 12 inches.

Applied rules:

• Three-number GCF: Same method extends to multiple numbers

• Tiling application: GCF gives optimal tile size for even coverage

• Dimension compatibility: Tile size must divide all dimensions evenly

Complete Summary: Greatest Common Factor (GCF)
GCF(a,b) = max{d : d|a and d|b}
GCF Definition
Key definitions:

Greatest Common Factor (GCF): The largest positive integer that divides evenly into two or more numbers without remainder

Factor: A number that divides another number evenly (a is a factor of b if b ÷ a is a whole number)

Common factor: A factor that two or more numbers share

Coprime numbers: Two numbers whose GCF is 1 (they share no common factors except 1)

Complete methodology:
  1. Identify the numbers: Determine which numbers to find GCF for
  2. Choose method: Select listing or prime factorization based on number size
  3. Execute method: Either list all factors or find prime factorizations
  4. Find commonality: Identify factors shared by all numbers
  5. Select maximum: Choose the largest common factor
  6. Verify result: Ensure GCF divides all original numbers evenly
Tip 1: For numbers under 20, listing method is usually faster.
Tip 2: For larger numbers, prime factorization method is more efficient.
Tip 3: Use divisibility rules to quickly identify factors.
Tip 4: Remember that 1 is always a common factor of any numbers.
Key applications: Simplifying fractions, finding common denominators, factoring polynomials, real-world equal groupings.
Mathematical significance: Forms the basis for understanding divisibility and number relationships.
Essential Rules:

• Size limitation: GCF is never larger than the smallest number

• Lower bound: GCF is always at least 1

• Self-property: GCF(a, a) = a

• Unity property: GCF(a, 1) = 1

• Prime property: If p is prime, GCF(p, a) = 1 or p (depending on whether p divides a)

• Relationship: GCF(a, b) × LCM(a, b) = a × b

Questions & Answers

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

Answer: Choose your method based on the numbers:

Use listing method when:

  • Numbers are small (≤ 20)
  • You can easily list all factors
  • Numbers have few factors
  • You're still learning the concept

Use prime factorization method when:

  • Numbers are large
  • Numbers have many factors
  • You need to find GCF 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: Can the GCF of two numbers ever be one of the original numbers? When does this happen?

Answer: Yes, the GCF can be one of the original numbers! This happens when one number is a factor of the other.

For example:

  • GCF(12, 24) = 12 (because 12 divides 24 evenly)
  • GCF(8, 16) = 8 (because 8 divides 16 evenly)
  • GCF(5, 15) = 5 (because 5 divides 15 evenly)

In general, if a divides b (meaning b ÷ a is a whole number), then GCF(a, b) = a.

The most extreme case is GCF(a, a) = a - any number's GCF with itself is the number itself.

On the other hand, if two numbers share no common factors except 1, their GCF is 1. Such numbers are called "coprime" or "relatively prime."

Example: GCF(7, 10) = 1 because 7 is prime and doesn't divide 10.

Question: How can I check if my GCF answer is correct? Is there a quick way to verify?

Answer: Yes, there are several ways to verify your GCF:

  1. Division check: Divide each original number by your GCF. The results should be whole numbers with no remainder.
  2. Factor check: Make sure your GCF is actually a factor of all original numbers.
  3. Commonality check: Verify that the quotients from step 1 share no common factors except 1.

Example: If you found GCF(36, 48) = 12:

  • 36 ÷ 12 = 3 ✓ (whole number)
  • 48 ÷ 12 = 4 ✓ (whole number)
  • GCF(3, 4) = 1 ✓ (no further common factors)

You can also use the relationship: GCF(a, b) × LCM(a, b) = a × b to double-check (though finding LCM might be extra work).

The division check is the quickest verification method!