Solved Exercises on Finding Prime Factors of Numbers in Grade 4

Master finding prime factors of numbers: factor trees, prime factorization, and factorization methods through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Simple Factor Tree
Exercise 1
Find the prime factors of 12 using a factor tree.
Definition:

Prime Factor: A factor that is a prime number. Factor Tree: A diagram showing how a number breaks down into its prime factors.

Factor Tree Method:
  1. Start with the number at the top
  2. Find two factors that multiply to give the number
  3. Continue breaking down composite factors until all are prime
  4. List all prime numbers at the bottom
12
4 × 3
2 × 2 × 3
Step 1: Start with 12

Write 12 at the top of the tree

Step 2: Break down 12

Find two factors: 12 = 4 × 3

Write 4 and 3 below 12

Step 3: Continue factoring

4 is not prime, so break it down: 4 = 2 × 2

2 and 3 are prime numbers

Step 4: List prime factors

All primes at the bottom: 2, 2, 3

2
2
3
2
2
3
Prime factors of 12: 2, 2, 3
Final answer:

Prime factors of 12: 2, 2, 3 (or 2² × 3)

Applied rules:

• Prime numbers: Have only 2 factors: 1 and themselves

• Composite numbers: Can be broken down further

• Factor tree: Shows the breakdown process visually

2 Prime Factorization
Exercise 2
Find the prime factors of 18.
Definition:

Prime Factorization: Writing a number as a product of its prime factors.

18
2 × 9
2 × 3 × 3
Step 1: Start with 18

Find two factors: 18 = 2 × 9

Step 2: Factor 9

9 = 3 × 3

3 is a prime number

Step 3: List all prime factors

From the bottom: 2, 3, 3

Step 4: Verify

Check: 2 × 3 × 3 = 18 ✓

2
3
3
2
3
3
Prime factors of 18: 2, 3, 3
Final answer:

Prime factors of 18: 2, 3, 3 (or 2 × 3²)

Applied rules:

• Prime factorization: Every number has a unique prime factorization

• Prime numbers: 2, 3, 5, 7, 11, 13, 17, 19...

• Verification: Multiply all prime factors to check

3 Larger Number Factorization
Exercise 3
Find the prime factors of 24.
Definition:

Prime Factorization: The process of expressing a number as a product of prime numbers.

24
4 × 6
2 × 2 × 2 × 3
Step 1: Start with 24

Find two factors: 24 = 4 × 6

Step 2: Factor 4

4 = 2 × 2

2 is prime

Step 3: Factor 6

6 = 2 × 3

2 and 3 are prime

Step 4: List all prime factors

From the bottom: 2, 2, 2, 3

Step 5: Verify

Check: 2 × 2 × 2 × 3 = 24 ✓

2
2
2
3
2
2
2
3
Prime factors of 24: 2, 2, 2, 3
Final answer:

Prime factors of 24: 2, 2, 2, 3 (or 2³ × 3)

Applied rules:

• Systematic approach: Break down composite numbers step by step

• Prime identification: Stop when all factors are prime

• Verification: Multiply all prime factors to confirm

Prime Factorization Rules and Methods
Number = Prime_1^{a} × Prime_2^{b} × Prime_3^{c}...
Prime Factorization
Rule 1
2 = 2
Smallest prime number
Rule 2
3 = 3
Next prime number
Rule 3
4 = 2 × 2
Composite number breakdown
Key definitions:

Prime Number: A number with exactly two factors: 1 and itself (2, 3, 5, 7, 11, 13...)

Composite Number: A number with more than two factors (4, 6, 8, 9, 10, 12...)

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

Factor Tree: A diagram showing how a number breaks down into its prime factors

Complete methodology:
  1. Start with the number: Write the number you want to factorize
  2. Find two factors: Look for any two numbers that multiply to give your number
  3. Check for primes: If a factor is prime, circle it
  4. Continue factoring: Break down composite factors until all are prime
  5. List prime factors: Write all the prime numbers you found
  6. Verify: Multiply all prime factors to ensure you get the original number
Tip 1: Start by dividing by the smallest prime (2).
Tip 2: If a number is even, it's divisible by 2.
Tip 3: If the sum of digits is divisible by 3, the number is divisible by 3.
Tip 4: Always verify by multiplying the prime factors.
Prime numbers to remember: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31...
Key insight: Every number has exactly one prime factorization (Fundamental Theorem of Arithmetic).
Prime Factorization Rules:

• Every number greater than 1 is either prime or composite

• Every composite number can be written as a product of primes

• The prime factorization is unique (order doesn't matter)

• 1 is neither prime nor composite

Solution: Exercises 4 to 5
4 Factorization with Powers
Exercise 4
Find the prime factors of 36 and write in exponential form.
Definition:

Exponential Form: Writing repeated prime factors using exponents (e.g., 2×2×2 = 2³).

36
4 × 9
2 × 2 × 3 × 3
Step 1: Start with 36

Find two factors: 36 = 4 × 9

Step 2: Factor 4

4 = 2 × 2

2 is prime

Step 3: Factor 9

9 = 3 × 3

3 is prime

Step 4: List all prime factors

From the bottom: 2, 2, 3, 3

Step 5: Write in exponential form

2 appears twice: 2²

3 appears twice: 3²

So: 36 = 2² × 3²

Step 6: Verify

Check: 2² × 3² = 4 × 9 = 36 ✓

2²
3²
2
2
3
3
Prime factors of 36: 2² × 3²
Final answer:

Prime factors of 36: 2² × 3²

Applied rules:

• Exponential notation: When the same prime appears multiple times

• Count occurrences: Count how many times each prime appears

• Verification: Multiply to ensure you get the original number

5 Large Number Factorization
Exercise 5
Find the prime factors of 60.
Definition:

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

60
6 × 10
2 × 3 × 2 × 5
Step 1: Start with 60

Find two factors: 60 = 6 × 10

Step 2: Factor 6

6 = 2 × 3

Both 2 and 3 are prime

Step 3: Factor 10

10 = 2 × 5

Both 2 and 5 are prime

Step 4: List all prime factors

From the bottom: 2, 3, 2, 5

Step 5: Organize and count

Prime factors: 2, 2, 3, 5

In exponential form: 2² × 3 × 5

Step 6: Verify

Check: 2² × 3 × 5 = 4 × 3 × 5 = 60 ✓

2
2
3
5
2
2
3
5
Prime factors of 60: 2, 2, 3, 5
Final answer:

Prime factors of 60: 2, 2, 3, 5 (or 2² × 3 × 5)

Applied rules:

• Systematic approach: Break down step by step

• Prime identification: Recognize when factors are prime

• Verification: Always check your answer by multiplying

Prime Factorization Complete Guide
n = p_1^{a} × p_2^{b} × p_3^{c}...
Prime Factorization Formula
Key definitions:

Prime Number: A number with exactly two factors: 1 and itself (2, 3, 5, 7, 11, 13...)

Composite Number: A number with more than two factors (4, 6, 8, 9, 10, 12...)

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

Factor Tree: A diagram showing how a number breaks down into its prime factors

Prime Factorization Methods:
  1. Factor Tree Method: Start with the number, break down into factors, continue until all are prime
  2. Division Method: Divide by smallest prime, continue dividing by primes
  3. Systematic Approach: Try divisors in order (2, 3, 5, 7, 11...)
  4. Verification: Multiply all prime factors to confirm original number
Tip 1: Start by dividing by 2 (if the number is even).
Tip 2: If the sum of digits is divisible by 3, the number is divisible by 3.
Tip 3: Numbers ending in 0 or 5 are divisible by 5.
Tip 4: Always verify by multiplying the prime factors together.
Tip 5: Keep a list of small prime numbers handy (2, 3, 5, 7, 11, 13...).
Prime factorization patterns: Every number has exactly one prime factorization (Fundamental Theorem of Arithmetic).
Key insight: Prime factorization helps understand number properties and relationships.
Important Properties:

• Every integer greater than 1 is either prime or composite

• Every composite number can be uniquely expressed as a product of primes

• Prime factorization is independent of the order in which factors are found

• 1 is neither prime nor composite

Prime Factorization Patterns
📊
Small Primes
2, 3, 5, 7, 11, 13, 17, 19, 23...
2
3
5
7
Factor Trees
Break down until all are prime
12
4 × 3
2 × 2 × 3
Exponential Form
2×2×3 = 2²×3
2²
3
Prime Factorization Steps
1
Start with number
2
Find factors
3
Break down composites
4
List primes
5
Verify
Prime Factorization Examples
12 = 2² × 3
18 = 2 × 3²
24 = 2³ × 3
36 = 2² × 3²
Remember: Every Number Has One Prime Factorization!

Questions & Answers

Question: How do I know if a number is prime or composite?

Answer: A prime number has exactly two factors: 1 and itself. A composite number has more than two factors:

  • Prime examples: 2 (factors: 1, 2), 3 (factors: 1, 3), 5 (factors: 1, 5)
  • Composite examples: 4 (factors: 1, 2, 4), 6 (factors: 1, 2, 3, 6), 8 (factors: 1, 2, 4, 8)

To check if a number is prime, try dividing it by small primes (2, 3, 5, 7...). If none divide evenly except 1 and itself, it's prime.

Remember: 2 is the only even prime number!

Question: Do I always have to start factoring with 2? Can I use other numbers?

Answer: You can start with any pair of factors! Here are different approaches:

  • Starting with 2: If the number is even, start with 2 (e.g., 24 = 2 × 12)
  • Other factors: You could start with 3, 4, 6, etc. (e.g., 24 = 3 × 8)
  • Whatever works: 24 = 4 × 6, then factor 4 and 6 further

No matter which pair you start with, you'll always end up with the same prime factors! For example, starting with 24 = 2 × 12 gives 2 × 2 × 2 × 3, and starting with 24 = 3 × 8 gives 3 × 2 × 2 × 2 - both equal 2³ × 3.

Starting with 2 is often easiest because it's the smallest prime!

Question: What should I do if I get stuck when trying to find prime factors?

Answer: Here's a systematic approach when you're stuck:

  1. Start with 2: If the number is even, divide by 2
  2. Try 3: If the sum of digits is divisible by 3, divide by 3
  3. Try 5: If it ends in 0 or 5, divide by 5
  4. Try 7, 11, 13: Continue with larger primes
  5. Make a factor tree: Draw it out to visualize the process
  6. Check your work: Multiply all prime factors to verify

If you're still stuck, try a smaller number first to practice the method. Remember, every number has a prime factorization!

Take it one step at a time and don't rush!