Solved Exercises on Greatest Common Factor (GCF) in Pre-algebra

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

Solution: Exercises 1 to 3
1 Listing Factors Method
Exercise 1
Find the GCF of 24 and 36 using the listing factors method.
Definition:

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

Listing Factors Method: Find all factors of each number and identify the largest common factor.

Listing Factors Method:
  1. List all factors of the first number
  2. List all factors of the second number
  3. Identify common factors
  4. Select the greatest common factor
Numbers
24 and 36
Common Factors
1, 2, 3, 4, 6, 12
GCF
12
Step 1: List factors of 24

1, 2, 3, 4, 6, 8, 12, 24

Step 2: List factors of 36

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

Step 3: Identify common factors

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

Step 4: Select the greatest

The largest common factor is 12

GCF(24, 36) = 12
Final answer:

The GCF of 24 and 36 is 12.

Applied rules:

• Factor Identification: Find all numbers that divide evenly

• Commonality: Look for factors shared by all numbers

• Greatest Value: Select the largest common factor

2 Prime Factorization Method
Exercise 2
Find the GCF of 48 and 72 using the prime factorization method.
Definition:

Prime Factorization Method: Find prime factorization of each number and multiply common prime factors with lowest exponents.

Numbers
48 and 72
Prime Factorizations
2⁴×3 and 2³×3²
GCF
2³×3 = 24
Step 1: Find prime factorization of 48

48 = 2⁴ × 3 = 2 × 2 × 2 × 2 × 3

Step 2: Find prime factorization of 72

72 = 2³ × 3² = 2 × 2 × 2 × 3 × 3

Step 3: Identify common prime factors

Common factors: 2 and 3

Lowest exponent of 2: min(4,3) = 3

Lowest exponent of 3: min(1,2) = 1

Step 4: Multiply common factors

GCF = 2³ × 3¹ = 8 × 3 = 24

GCF(48, 72) = 24
Final answer:

The GCF of 48 and 72 is 24.

Applied rules:

• Prime Factorization: Express each number as product of primes

• Common Factors: Identify primes present in all factorizations

• Lowest Exponents: Use minimum exponent for each common prime

3 Three Numbers GCF
Exercise 3
Find the GCF of 30, 45, and 75 using the prime factorization method.
Definition:

GCF of Multiple Numbers: The largest positive integer that divides all numbers without remainder.

Numbers
30, 45, 75
Prime Factorizations
2×3×5, 3²×5, 3×5²
GCF
3×5 = 15
Step 1: Find prime factorization of 30

30 = 2 × 3 × 5

Step 2: Find prime factorization of 45

45 = 3² × 5 = 3 × 3 × 5

Step 3: Find prime factorization of 75

75 = 3 × 5² = 3 × 5 × 5

Step 4: Identify common prime factors

Prime factor 2: Present in 30 only → Not common

Prime factor 3: Present in all three → Take lowest exponent (1)

Prime factor 5: Present in all three → Take lowest exponent (1)

Step 5: Multiply common factors

GCF = 3¹ × 5¹ = 3 × 5 = 15

GCF(30, 45, 75) = 15
Final answer:

The GCF of 30, 45, and 75 is 15.

Applied rules:

• Multiple Numbers: All prime factors must be present in every number

• Lowest Exponents: Use minimum exponent across all factorizations

• Common Requirement: A prime must appear in ALL numbers to be included

GCF Methods & Techniques
GCF(a,b) = Product of common prime factors with lowest exponents
GCF Formula
Method 1
Listing Factors
Best for small numbers
Method 2
Prime Factorization
Most reliable method
Method 3
Euclidean Algorithm
Efficient for large numbers
Key definitions:

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

Common Factor: A number that divides each of the given numbers.

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

Euclidean Algorithm: A method for finding GCF using repeated division.

Complete methodology:
  1. Listing Factors: For small numbers, list all factors and find the largest common one
  2. Prime Factorization: Find prime factors of each number and multiply common factors with lowest exponents
  3. Euclidean Algorithm: For large numbers, use repeated division
  4. Verification: Check that GCF divides all original numbers evenly
Tip 1: Use prime factorization method for most problems - it's most reliable
Tip 2: For GCF of multiple numbers, all primes must appear in every factorization
Tip 3: Always verify by dividing original numbers by GCF to check for whole numbers
Tip 4: Remember that 1 is always a common factor
Common errors: Including non-common factors, using highest instead of lowest exponents, forgetting that GCF must divide all numbers evenly.
Key properties: GCF(a,b) ≤ min(a,b), GCF of coprime numbers is 1, GCF(a,a) = a.
Essential rules:

• Common Requirement: Only include primes that appear in ALL factorizations

• Lowest Exponents: Use minimum exponent for each common prime

• Verification: GCF must divide each original number without remainder

• Maximum Value: GCF cannot exceed the smallest number

Solution: Exercises 4 to 5
4 Euclidean Algorithm
Exercise 4
Find the GCF of 84 and 56 using the Euclidean algorithm.
Definition:

Euclidean Algorithm: A method for finding GCF using repeated division: GCF(a,b) = GCF(b, a mod b) until remainder is 0.

Numbers
84 and 56
Algorithm Steps
GCF(84,56)→GCF(56,28)→GCF(28,0)
GCF
28
Step 1: Apply Euclidean algorithm

84 ÷ 56 = 1 remainder 28

So GCF(84, 56) = GCF(56, 28)

Step 2: Continue algorithm

56 ÷ 28 = 2 remainder 0

So GCF(56, 28) = GCF(28, 0)

Step 3: When remainder is 0

When the second number is 0, the first number is the GCF

So GCF(28, 0) = 28

Step 4: Verification

84 ÷ 28 = 3 (whole number)

56 ÷ 28 = 2 (whole number)

✓ GCF divides both numbers evenly

GCF(84, 56) = 28
Final answer:

The GCF of 84 and 56 is 28.

Applied rules:

• Euclidean Algorithm: GCF(a,b) = GCF(b, a mod b)

• Termination: When remainder is 0, the first number is GCF

• Verification: GCF must divide both original numbers

5 Application Problem
Exercise 5
A teacher has 36 red markers, 48 blue markers, and 60 green markers. She wants to create identical sets of markers for her students, with no markers left over. What is the maximum number of sets she can make? How many markers of each color will be in each set?
Definition:

Application Problem: Real-world scenario requiring GCF to find maximum equal groups.

Markers
36, 48, 60
GCF
12
Per Set
3R, 4B, 5G
Step 1: Identify the problem

We need to find the largest number that divides 36, 48, and 60 evenly

This is the GCF of 36, 48, and 60

Step 2: Find GCF of 36, 48, and 60

Prime factorization of 36: 2² × 3²

Prime factorization of 48: 2⁴ × 3

Prime factorization of 60: 2² × 3 × 5

Step 3: Identify common factors

Common prime factors: 2 and 3

Lowest exponent of 2: min(2,4,2) = 2

Lowest exponent of 3: min(2,1,1) = 1

Step 4: Calculate GCF

GCF = 2² × 3¹ = 4 × 3 = 12

Step 5: Calculate markers per set

Red markers per set: 36 ÷ 12 = 3

Blue markers per set: 48 ÷ 12 = 4

Green markers per set: 60 ÷ 12 = 5

Step 6: Verify

Total sets: 12

Total markers: 12×(3+4+5) = 12×12 = 144

Original markers: 36+48+60 = 144 ✓

12 sets with 3R, 4B, 5G each
Final answer:

The teacher can make 12 identical sets. Each set will contain 3 red markers, 4 blue markers, and 5 green markers.

Applied rules:

• Real-world Application: GCF helps find maximum equal groups

• Division Principle: Divide total by GCF to find items per group

• Verification: Check that division results in whole numbers

GCF Applications & Advanced Techniques
GCF(a,b) = ∏(pᵢ^(min(e₁,e₂)))
GCF Formula
Key definitions:

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

Highest Common Factor (HCF): Alternative name for GCF, used in some regions.

Coprime Numbers: Numbers whose GCF is 1 (no common factors other than 1).

Relatively Prime: Another term for coprime numbers.

Complete methodology:
  1. For Small Numbers: Use listing factors method
  2. For Medium Numbers: Use prime factorization method
  3. For Large Numbers: Use Euclidean algorithm
  4. For Multiple Numbers: Apply method iteratively
Tip 1: GCF is essential for reducing fractions to lowest terms
Tip 2: If GCF(a,b) = 1, then a and b are coprime
Tip 3: GCF(a,b) × LCM(a,b) = a × b
Tip 4: Always verify by ensuring GCF divides all original numbers evenly
Common errors: Including non-common factors, using highest instead of lowest exponents, forgetting that GCF must divide all numbers evenly.
Applications: Fraction reduction, common denominators, equal grouping problems, cryptography.
Essential rules:

• Common Requirement: Only include primes that appear in ALL factorizations

• Lowest Exponents: Use minimum exponent for each common prime

• Verification: GCF must divide each original number without remainder

• Maximum Value: GCF cannot exceed the smallest number

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

Questions & Answers

Question: I'm confused about when to use the different methods for finding GCF. How do I choose the best one?

Answer: Here's a guide for choosing the best GCF method:

  • Listing Factors (Small numbers, ≤ 20): Quick and visual for small numbers
  • Prime Factorization (Medium numbers, 20-200): Most reliable method, works for any numbers
  • Euclidean Algorithm (Large numbers, > 200): Most efficient for large numbers

For most pre-algebra problems, prime factorization is recommended because it's systematic and works well for medium-sized numbers. It also reinforces understanding of prime numbers and factorization, which are important concepts.

Question: What happens when two numbers have no common factors besides 1? What's their GCF?

Answer: When two numbers have no common factors besides 1, they are called coprime or relatively prime. In this case, their GCF is 1.

Examples:

  • GCF(7, 10) = 1 (factors of 7: {1,7}, factors of 10: {1,2,5,10})
  • GCF(8, 15) = 1 (factors of 8: {1,2,4,8}, factors of 15: {1,3,5,15})
  • GCF(11, 13) = 1 (both are prime numbers)

This is important because coprime numbers have special properties in number theory and are often used in cryptography.

Question: How is GCF related to reducing fractions to lowest terms?

Answer: GCF is fundamental to reducing fractions to lowest terms:

  1. Find GCF: Calculate the GCF of the numerator and denominator
  2. Divide Both: Divide both the numerator and denominator by the GCF
  3. Result: The fraction is now in its simplest form

Example: Reduce 24/36

  • GCF(24, 36) = 12
  • Numerator: 24 ÷ 12 = 2
  • Denominator: 36 ÷ 12 = 3
  • Result: 24/36 = 2/3

This process ensures the fraction is in its simplest form, where the numerator and denominator are coprime.