Solved Exercises on Prime Factorization in Grade 6

Master prime factorization: factor trees, exponential notation, and applications through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Factor Tree Method
Exercise 1
Find the prime factorization of 36 using a factor tree. Express your answer using exponential notation.
Definition:

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

Factor tree method:
  1. Start with the number: Write the original number at the top
  2. Find a factor pair: Break it into two factors
  3. Continue factoring: Factor any composite numbers
  4. Stop at primes: When all branches end in prime numbers
  5. Write in exponential form: Group repeated factors
Factor Tree for 36:
36
├── 4
│ ├── 2
│ └── 2
└── 9
├── 3
└── 3
Step 1: Start with 36

Find a factor pair: 36 = 4 × 9

Step 2: Factor 4 and 9

4 = 2 × 2 (both 2s are prime)

9 = 3 × 3 (both 3s are prime)

Step 3: Identify all prime factors

From the tree: 2, 2, 3, 3

Step 4: Write in exponential notation

2 appears twice: 2²

3 appears twice: 3²

Prime factorization: 2² × 3²

Step 5: Verify the result

2² × 3² = 4 × 9 = 36 ✓

36 = 2² × 3²
Final answer:

The prime factorization of 36 is 2² × 3².

Applied rules:

• Factor tree principle: Each composite number splits into factors

• Prime termination: Stop factoring when reaching prime numbers

• Exponential notation: Count repeated factors using exponents

2 Repeated Division Method
Exercise 2
Find the prime factorization of 84 using the repeated division method. Show all division steps.
Definition:

Repeated division method: Continuously divide by the smallest prime factors

Repeated Division for 84:
84 ÷ 2 = 42
42 ÷ 2 = 21
21 ÷ 3 = 7
7 ÷ 7 = 1
Step 1: Divide by the smallest prime factor (2)

84 ÷ 2 = 42

Step 2: Continue dividing by 2

42 ÷ 2 = 21

Step 3: Move to next smallest prime (3)

21 ÷ 3 = 7

Step 4: Finish with the remaining prime

7 ÷ 7 = 1

Step 5: Collect all prime factors

Prime factors from divisions: 2, 2, 3, 7

Step 6: Write in exponential notation

2² × 3¹ × 7¹ = 2² × 3 × 7

84 = 2² × 3 × 7
Final answer:

The prime factorization of 84 is 2² × 3 × 7.

Applied rules:

• Division order: Start with smallest primes (2, 3, 5, 7, 11...)

• Continue until 1: Keep dividing until result is 1

• Record divisors: Each divisor is a prime factor

3 Verification and Applications
Exercise 3
Find the prime factorization of 60 and 75. Then use these to find the GCF and LCM of 60 and 75.
Definition:

Applications: Prime factorization helps find GCF and LCM efficiently

Prime Factorization of 60:

60 ÷ 2 = 30

30 ÷ 2 = 15

15 ÷ 3 = 5

5 ÷ 5 = 1

So 60 = 2² × 3 × 5

Prime Factorization of 75:

75 ÷ 3 = 25

25 ÷ 5 = 5

5 ÷ 5 = 1

So 75 = 3 × 5²

Step 1: Find prime factorization of 60

60 = 2² × 3 × 5

Step 2: Find prime factorization of 75

75 = 3 × 5²

Step 3: Find GCF using prime factorizations

60 = 2² × 3¹ × 5¹

75 = 3¹ × 5²

Common primes: 3 and 5

Take lowest powers: 3¹ × 5¹ = 15

Step 4: Find LCM using prime factorizations

Take highest power of each prime: 2² × 3¹ × 5² = 4 × 3 × 25 = 300

Step 5: Verify results

GCF verification: 60 ÷ 15 = 4, 75 ÷ 15 = 5 (4 and 5 share no common factors)

LCM verification: 300 ÷ 60 = 5, 300 ÷ 75 = 4

GCF(60,75) = 15; LCM(60,75) = 300
Final answer:

Prime factorization of 60: 2² × 3 × 5

Prime factorization of 75: 3 × 5²

GCF of 60 and 75: 15

LCM of 60 and 75: 300

Applied rules:

• GCF method: Take lowest power of common prime factors

• LCM method: Take highest power of all prime factors

• Verification: GCF × LCM = original numbers' product

Prime Factorization Methods and Properties
n = p₁^a₁ × p₂^a₂ × ... × pₖ^aₖ
Fundamental Theorem
Method 1
Factor Tree
Visual branching approach
Method 2
Repeated Division
Systematic division approach
Method 3
Exponential Form
Compact notation
Key definitions:

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

Fundamental theorem of arithmetic: Every integer greater than 1 has a unique prime factorization

Factorization Process:
  1. Identify the number: Determine what number to factorize
  2. Choose method: Select factor tree or repeated division
  3. Begin factoring: Start with smallest prime factors
  4. Continue systematically: Factor each composite result
  5. Stop at primes: When all factors are prime
  6. Express compactly: Use exponential notation for repeated factors
Tip 1: Always start with the smallest prime (2), then try 3, 5, 7, 11...
Tip 2: Use divisibility rules to quickly identify factors.
Tip 3: Remember that 2 is the only even prime number.
Tip 4: Check your work by multiplying prime factors to get original number.
Common errors: Forgetting to continue factoring, missing prime factors, incorrect exponential notation.
Success habits: Working systematically, checking work, using divisibility rules.
Prime Numbers (First 25):

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

Remember: 1 is neither prime nor composite

Solution: Exercises 4 to 5
4 Large Number Factorization
Exercise 4
Find the prime factorization of 252. Use the most efficient method and verify your answer.
Definition:

Large number factorization: Using systematic approaches for numbers with multiple prime factors

Prime Factorization of 252:

252 ÷ 2 = 126

126 ÷ 2 = 63

63 ÷ 3 = 21

21 ÷ 3 = 7

7 ÷ 7 = 1

Step 1: Divide by 2 (smallest prime)

252 is even, so divisible by 2

252 ÷ 2 = 126

Step 2: Continue dividing by 2

126 is even, so divisible by 2

126 ÷ 2 = 63

Step 3: Move to next prime (3)

Sum of digits of 63: 6 + 3 = 9, divisible by 3

63 ÷ 3 = 21

Step 4: Continue with 3

Sum of digits of 21: 2 + 1 = 3, divisible by 3

21 ÷ 3 = 7

Step 5: Finish with prime 7

7 is prime, so 7 ÷ 7 = 1

Step 6: Write prime factorization

Prime factors: 2, 2, 3, 3, 7

In exponential form: 2² × 3² × 7

Step 7: Verify the answer

2² × 3² × 7 = 4 × 9 × 7 = 36 × 7 = 252 ✓

252 = 2² × 3² × 7
Final answer:

The prime factorization of 252 is 2² × 3² × 7.

Applied rules:

• Divisibility rules: Use rules to identify factors quickly

• Systematic approach: Work from smallest to largest primes

• Verification: Always check by multiplying factors

5 Real-World Application
Exercise 5
A rectangular garden has an area of 120 square feet. If the length and width must be whole numbers, what are all possible dimensions? Use prime factorization to find the answer.
Definition:

Real-world application: Using prime factorization to find all factor pairs of a number

Finding All Factors of 120:

Prime factorization: 120 = 2³ × 3 × 5

Using prime factorization to find all factors:

Possible combinations of prime powers:

2⁰, 2¹, 2², 2³ and 3⁰, 3¹ and 5⁰, 5¹

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 × 3 × 5

Step 2: Find all factors using prime factorization

Factors are formed by taking 0-3 powers of 2, 0-1 power of 3, 0-1 power of 5

All factors: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120

Step 3: Pair factors to get dimensions

1 × 120 = 120

2 × 60 = 120

3 × 40 = 120

4 × 30 = 120

5 × 24 = 120

6 × 20 = 120

8 × 15 = 120

10 × 12 = 120

Step 4: List all possible dimensions

(1, 120), (2, 60), (3, 40), (4, 30), (5, 24), (6, 20), (8, 15), (10, 12)

Step 5: Verify each pair

Each pair multiplies to 120: 10 × 12 = 120 ✓

Possible dimensions: (1,120), (2,60), (3,40), (4,30), (5,24), (6,20), (8,15), (10,12)
Final answer:

The possible dimensions for the garden are: 1×120, 2×60, 3×40, 4×30, 5×24, 6×20, 8×15, and 10×12 feet.

Applied rules:

• Factor pair principle: Length × Width = Area

• Prime factorization utility: Efficiently finds all factors

• Real-world constraints: Both dimensions must be positive integers

Complete Summary: Prime Factorization
Every Integer > 1 = Unique Prime Product
Fundamental Theorem
Key definitions:

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

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

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

Fundamental theorem of arithmetic: Every integer greater than 1 can be uniquely expressed as a product of primes

Complete methodology:
  1. Identify the number: Confirm it's composite (not prime)
  2. Choose approach: Factor tree for visual learners or repeated division for systematic learners
  3. Begin with smallest primes: Start with 2, then 3, 5, 7, 11, etc.
  4. Continue factoring: Keep dividing composite results
  5. Stop at primes: When all factors are prime numbers
  6. Express compactly: Group repeated primes using exponents
  7. Verify result: Multiply factors to confirm original number
Tip 1: Memorize the first 10 prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.
Tip 2: Use divisibility rules to quickly identify which primes divide your number.
Tip 3: You only need to check prime divisors up to the square root of the number.
Tip 4: Always verify your answer by multiplying the prime factors.
Key applications: Finding GCF, LCM, simplifying fractions, cryptography, number theory.
Mathematical significance: Forms the foundation for understanding number properties and relationships.
Essential Rules:

• Uniqueness: Prime factorization is unique for each number (ignoring order)

• Systematic approach: Always start with the smallest prime factor

• Termination: Process ends when all factors are prime

• Exponential notation: Write repeated factors as powers (e.g., 2×2×2 = 2³)

• Verification: Multiply prime factors to confirm original number

Questions & Answers

Question: I'm not sure which method is better - factor trees or repeated division. How do I choose?

Answer: Both methods lead to the same result - choose based on your preference:

Factor tree method is better if:

  • You're a visual learner who likes seeing the branching structure
  • You prefer to think of factor pairs rather than sequential division
  • You want to see all the steps laid out spatially

Repeated division method is better if:

  • You prefer a systematic, step-by-step approach
  • You like working with sequential calculations
  • You want a more compact written solution

For smaller numbers (under 50), either method works well. For larger numbers, repeated division is often more efficient as it's more systematic.

Try both methods with the same number to see which feels more natural to you!

Question: Why do we need to know prime factorization? When will I ever use this in real life?

Answer: Prime factorization has many practical applications:

  • Finding GCF/LCM: Useful for adding/subtracting fractions, scheduling problems
  • Simplifying fractions: Cancel common prime factors in numerator and denominator
  • Real-world problems: Arranging items in rectangles, packaging, organizing
  • Cryptography: Modern security systems rely on difficulty of factoring large primes
  • Computer science: Algorithm design and optimization

Example: If you're tiling a floor and need to find the largest square tile that fits perfectly, you'd use GCF, which requires prime factorization.

Even if you don't directly factor numbers as an adult, the logical thinking and systematic approach you develop through prime factorization benefits you in problem-solving across all areas of life.

It's also foundational for higher mathematics!

Question: How can I quickly tell if a number is prime? Do I have to try dividing by every number?

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 97 is prime:

  • Is 97 even? No, so not divisible by 2
  • Sum of digits: 9 + 7 = 16, not divisible by 3
  • Doesn't end in 0 or 5, so not divisible by 5
  • √97 ≈ 9.8, so only check primes up to 9: just 7
  • 97 ÷ 7 = 13.86... (not whole), so not divisible by 7

Since we've checked all primes up to √97 and none divide evenly, 97 is prime!

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