Solved Exercises on Prime Factorization in Pre-algebra

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

Solution: Exercises 1 to 3
1 Basic Factor Tree
Exercise 1
Find the prime factorization of 24 using a factor tree. Express your answer in exponential form.
Definition:

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

Factor Tree: A visual representation showing how a number breaks down into prime factors.

Exponential Form: Using exponents to show repeated multiplication of the same prime.

Factor Tree Method:
  1. Start with the given number at the top
  2. Find two factors (not necessarily prime) and branch them out
  3. Continue factoring until all branches end in prime numbers
  4. Write the prime factors as a product
  5. Express in exponential form if there are repeated factors
Start
24
Factor Tree
2³ × 3
Result
24 = 2³ × 3
Step 1: Start with 24

24 ÷ 2 = 12 → 2 is a factor

Step 2: Factor 12

12 ÷ 2 = 6 → 2 is a factor again

Step 3: Factor 6

6 ÷ 2 = 3 → 2 is a factor again

Step 4: Factor 3

3 is prime → 3 is our final factor

Step 5: Write in exponential form

24 = 2 × 2 × 2 × 3 = 2³ × 3

24 = 2³ × 3
Final answer:

The prime factorization of 24 is 2³ × 3.

Applied rules:

• Systematic Division: Always divide by smallest available prime

• Prime Recognition: Stop when reaching a prime number

• Exponential Form: Group repeated prime factors

2 Division Method
Exercise 2
Find the prime factorization of 60 using the division method. Express your answer in exponential form.
Definition:

Division Method: Repeatedly dividing the number by the smallest prime factor until reaching 1.

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

Number
60
Division Steps
2² × 3 × 5
Result
60 = 2² × 3 × 5
Step 1: Start with 60

60 ÷ 2 = 30 → Record factor 2

Step 2: Factor 30

30 ÷ 2 = 15 → Record factor 2

Step 3: Factor 15

15 ÷ 3 = 5 → Record factor 3

Step 4: Factor 5

5 ÷ 5 = 1 → Record factor 5

Step 5: Write in exponential form

60 = 2 × 2 × 3 × 5 = 2² × 3 × 5

60 = 2² × 3 × 5
Final answer:

The prime factorization of 60 is 2² × 3 × 5.

Applied rules:

• Smallest Prime First: Always divide by the smallest prime factor

• Continue Until 1: Keep dividing until the quotient is 1

• Record All Factors: Keep track of each prime divisor

3 Larger Number Factorization
Exercise 3
Find the prime factorization of 144 using the division method. Express your answer in exponential form.
Definition:

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

Exponential Form: Using exponents to show repeated multiplication of the same prime.

Number
144
Division Steps
2⁴ × 3²
Result
144 = 2⁴ × 3²
Step 1: Start with 144

144 ÷ 2 = 72 → Record factor 2

Step 2: Factor 72

72 ÷ 2 = 36 → Record factor 2

Step 3: Factor 36

36 ÷ 2 = 18 → Record factor 2

Step 4: Factor 18

18 ÷ 2 = 9 → Record factor 2

Step 5: Factor 9

9 ÷ 3 = 3 → Record factor 3

Step 6: Factor 3

3 ÷ 3 = 1 → Record factor 3

Step 7: Write in exponential form

144 = 2 × 2 × 2 × 2 × 3 × 3 = 2⁴ × 3²

144 = 2⁴ × 3²
Final answer:

The prime factorization of 144 is 2⁴ × 3².

Applied rules:

• Systematic Approach: Always divide by smallest prime factor

• Complete Process: Continue until quotient is 1

• Exponential Grouping: Count repetitions and express as powers

Prime Factorization Summary
n = p₁^a × p₂^b × p₃^c...
Prime Factorization
Method 1
Factor Tree
Visual breakdown of factors
Method 2
Division Method
Systematic division by primes
Exponential Form
2⁴ × 3²
Group repeated factors
Key definitions:

Prime Number: A natural number greater than 1 that has exactly two distinct positive divisors: 1 and itself.

Composite Number: A natural number greater than 1 that has more than two positive divisors.

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

Exponential Form: Using exponents to represent repeated multiplication of the same prime.

Complete methodology:
  1. Choose Method: Factor tree for visualization or division method for systematic approach
  2. Start Factoring: Use smallest prime factor (2, 3, 5, 7, 11, ...)
  3. Continue Process: Factor until all numbers are prime
  4. Express Result: Write as product of primes
  5. Simplify: Convert to exponential form
Tip 1: Always start with the smallest prime factor (2, then 3, then 5, etc.)
Tip 2: Use divisibility rules to quickly identify factors
Tip 3: Stop when you reach a prime number
Tip 4: Verify by multiplying factors back together
Common errors: Forgetting to continue factoring until all numbers are prime, miscounting repeated factors, not converting to exponential form.
Key properties: Every composite number has a unique prime factorization (Fundamental Theorem of Arithmetic).
Essential rules:

• Smallest Prime First: Always divide by the smallest available prime

• Stop at Primes: Don't factor prime numbers further

• Record All Factors: Keep track of every prime divisor

• Exponential Form: Group identical factors with exponents

• Verification: Multiply factors to check accuracy

Solution: Exercises 4 to 5
4 Complex Factorization
Exercise 4
Find the prime factorization of 360. Express your answer in exponential form and verify your result.
Definition:

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

Verification: Checking the result by multiplying factors back together.

Start
360
Factorization
2³ × 3² × 5
Verification
8 × 9 × 5 = 360
Step 1: Start with 360

360 ÷ 2 = 180 → Record factor 2

Step 2: Factor 180

180 ÷ 2 = 90 → Record factor 2

Step 3: Factor 90

90 ÷ 2 = 45 → Record factor 2

Step 4: Factor 45

45 ÷ 3 = 15 → Record factor 3

Step 5: Factor 15

15 ÷ 3 = 5 → Record factor 3

Step 6: Factor 5

5 ÷ 5 = 1 → Record factor 5

Step 7: Write in exponential form

360 = 2 × 2 × 2 × 3 × 3 × 5 = 2³ × 3² × 5

Step 8: Verification

2³ × 3² × 5 = 8 × 9 × 5 = 72 × 5 = 360 ✓

360 = 2³ × 3² × 5
Final answer:

The prime factorization of 360 is 2³ × 3² × 5.

Applied rules:

• Systematic Division: Always divide by smallest prime factor

• Complete Process: Continue until quotient is 1

• Verification: Multiply factors back to check

5 Application Problem
Exercise 5
A rectangular room has an area of 84 square feet. Find the prime factorization of 84. Then, determine all possible whole number dimensions of the room where both length and width are expressed as products of prime factors.
Definition:

Area of Rectangle: Length × Width = Area

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

Area
84
Prime Factors
2² × 3 × 7
Dimensions
(2²×3)×7, 2²×(3×7), etc.
Step 1: Find prime factorization of 84

84 ÷ 2 = 42 → Record factor 2

42 ÷ 2 = 21 → Record factor 2

21 ÷ 3 = 7 → Record factor 3

7 ÷ 7 = 1 → Record factor 7

So 84 = 2² × 3 × 7

Step 2: Find all factor pairs of 84

Using prime factors: 2² × 3 × 7 = 4 × 3 × 7

All factor pairs: 1×84, 2×42, 3×28, 4×21, 6×14, 7×12

Step 3: Express each dimension as products of primes

1×84: 1 × (2² × 3 × 7)

2×42: 2 × (2 × 3 × 7)

3×28: 3 × (2² × 7)

4×21: (2²) × (3 × 7)

6×14: (2 × 3) × (2 × 7)

7×12: 7 × (2² × 3)

Step 4: List all possible dimensions

1×84, 2×42, 3×28, 4×21, 6×14, 7×12

With prime factorization: (1)×(2²×3×7), (2)×(2×3×7), (3)×(2²×7), (2²)×(3×7), (2×3)×(2×7), (7)×(2²×3)

84 = 2² × 3 × 7; Dimensions: 1×84, 2×42, 3×28, 4×21, 6×14, 7×12
Final answer:

The prime factorization of 84 is 2² × 3 × 7. The possible dimensions are: 1×84, 2×42, 3×28, 4×21, 6×14, 7×12.

Applied rules:

• Area Formula: Length × Width = Area

• Systematic Listing: Find all factor pairs

• Prime Factorization: Express dimensions using prime factors

Prime Factorization Techniques & Applications
n = p₁^a × p₂^b × p₃^c...
Prime Factorization
Key definitions:

Prime Number: A natural number greater than 1 that has exactly two distinct positive divisors: 1 and itself.

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

Exponential Form: Using exponents to represent repeated multiplication of the same prime.

Fundamental Theorem of Arithmetic: Every integer greater than 1 has a unique prime factorization (up to order of factors).

Complete methodology:
  1. Factor Tree Method: Visual approach starting with the number and branching to factors
  2. Division Method: Systematic division by smallest prime factors
  3. Verification: Multiply factors to ensure original number
  4. Application: Use factorization for LCM, GCD, simplification
Tip 1: Memorize primes up to 30: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29
Tip 2: Use divisibility rules: 2 (even), 3 (sum of digits), 5 (ends in 0 or 5)
Tip 3: Group identical prime factors using exponents
Tip 4: Always verify by multiplying factors back together
Common errors: Stopping too early before reaching all primes, miscounting repeated factors, not converting to exponential form.
Applications: Finding GCD/LCM, simplifying fractions, solving word problems, cryptography.
Essential rules:

• Smallest Prime First: Always divide by the smallest available prime factor

• Stop at Primes: Don't factor prime numbers further

• Record All Factors: Keep track of every prime divisor

• Exponential Form: Group identical factors with exponents

• Verification: Multiply factors to check accuracy

Questions & Answers

Question: I get confused about when to stop factoring. How do I know if a number is prime?

Answer: Great question! Here's how to identify when to stop factoring:

  • Memorize small primes: Know that 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31 are prime
  • Check divisibility: Test if the number is divisible by primes up to √n
  • Quick checks: Even numbers > 2 are composite; numbers ending in 5 > 5 are composite
  • Pattern recognition: Practice will help you recognize primes faster

For example, when factoring 97: It's not even, doesn't end in 5, sum of digits (9+7=16) isn't divisible by 3, and √97 ≈ 9.8. Since it's not divisible by 2, 3, 5, or 7, it's prime and you stop.

Question: Is there more than one way to find the prime factorization of a number? Does the order matter?

Answer: Yes, there are multiple approaches to find prime factorization:

  • Factor Tree: Visual approach starting with any factor pair
  • Division Method: Systematic division by smallest primes
  • Grouping Strategy: Factor out obvious patterns first

However, the Fundamental Theorem of Arithmetic guarantees that the prime factorization is unique regardless of the path taken. The order of factors doesn't matter due to the commutative property of multiplication. For example, 60 = 2×2×3×5 = 2×3×2×5 = 5×2×2×3, but the prime factorization is always 2²×3×5.

Question: How do I convert from expanded form to exponential form efficiently?

Answer: Here's an efficient method to convert to exponential form:

  1. List all prime factors: Write them in order: 2×2×2×3×3×5
  2. Group identical factors: (2×2×2)×(3×3)×5
  3. Count occurrences: 2 appears 3 times, 3 appears 2 times, 5 appears 1 time
  4. Write with exponents: 2³×3²×5¹ (or just 2³×3²×5 since ⁵¹ = 5)

Tip: Organize your work systematically - write factors in ascending order as you find them, which makes grouping much easier. Also, remember that any number to the power of 1 is just the number itself (so 5¹ = 5).