Solved Exercises on Prime and Composite Numbers in Grade 4

Master prime and composite numbers: identification, factorization, and number classification through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Identifying Prime Numbers
Exercise 1
Which of these numbers are prime? Circle them: 2, 4, 5, 6, 7, 8, 9, 11, 12, 13
Definition:

Prime Number: A number greater than 1 that has exactly two factors: 1 and itself

Composite Number: A number greater than 1 that has more than two factors

Identification method:
  1. List all factors of each number
  2. Count how many factors exist
  3. If exactly 2 factors → Prime
  4. If more than 2 factors → Composite
Step 1: Analyze each number
2: 1, 2 → PRIME (2 factors)
4: 1, 2, 4 → COMPOSITE (3 factors)
5: 1, 5 → PRIME (2 factors)
6: 1, 2, 3, 6 → COMPOSITE (4 factors)
7: 1, 7 → PRIME (2 factors)
8: 1, 2, 4, 8 → COMPOSITE (4 factors)
9: 1, 3, 9 → COMPOSITE (3 factors)
11: 1, 11 → PRIME (2 factors)
12: 1, 2, 3, 4, 6, 12 → COMPOSITE (6 factors)
13: 1, 13 → PRIME (2 factors)
Step 2: Circle the prime numbers

Prime numbers in the list: 2, 5, 7, 11, 13

Step 3: Verify results

Each circled number has exactly 2 factors: 1 and itself

Prime numbers: 2, 5, 7, 11, 13
Final answer:

Prime numbers: 2, 5, 7, 11, 13

Applied rules:

• Prime definition: Exactly 2 factors (1 and itself)

• Composite definition: More than 2 factors

• Number 1: Neither prime nor composite (has only 1 factor)

2
4
5
6
7
8
9
11
12
13
2 Finding All Factors
Exercise 2
List all factors of 24. Then determine if 24 is prime or composite.
Definition:

Factor: A number that divides another number evenly with no remainder

Factor pairs: Two numbers that multiply to give the original number

Step 1: Find factor pairs systematically

Start with 1 and work up: 1×24, 2×12, 3×8, 4×6

Step 2: List all factors
1, 2, 3, 4, 6, 8, 12, 24
Step 3: Count the factors

24 has 8 factors total

Step 4: Determine if prime or composite

Since 24 has more than 2 factors, it is composite

Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Final answer:

Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24

24 is a composite number

Applied rules:

• Systematic approach: Check divisibility starting from 1

• Factor pairs: If a×b = n, then both a and b are factors of n

• Prime vs composite: Count factors to determine classification

Factor Pairs of 24
1 × 24 = 24
Factors: 1, 24
2 × 12 = 24
Factors: 2, 12
3 × 8 = 24
Factors: 3, 8
4 × 6 = 24
Factors: 4, 6
3 Prime Factorization
Exercise 3
Find the prime factorization of 36. Show your work using a factor tree.
Definition:

Prime Factorization: Writing a composite number as a product of prime numbers

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

Step 1: Start with the number 36

Find any two factors of 36: 36 = 4 × 9

Step 2: Break down non-prime factors

4 = 2 × 2 (both 2s are prime)

9 = 3 × 3 (both 3s are prime)

Step 3: Write the prime factorization

36 = 2 × 2 × 3 × 3 = 2² × 3²

Step 4: Verify the result

2² × 3² = 4 × 9 = 36 ✓

36 = 2² × 3²
Final answer:

Prime factorization of 36: 2² × 3²

Applied rules:

• Factor tree method: Break down composite numbers into smaller factors

• Continue until all factors are prime: Stop when only prime numbers remain

• Exponential notation: Write repeated prime factors using exponents

Factor Tree for 36
36
4
9
2
2
3
3
36 = 2² × 3²
Prime and Composite Numbers Rules and Methods
Prime: Exactly 2 factors
Prime Number Definition
Prime Numbers
2, 3, 5, 7, 11, 13, 17, 19, 23, 29...
Only factors: 1 and itself
Composite Numbers
4, 6, 8, 9, 10, 12, 14, 15, 16, 18...
More than 2 factors
Special Case
1
Neither prime nor composite
Key definitions:

Prime Number: A whole number greater than 1 that has exactly two factors: 1 and itself

Composite Number: A whole number greater than 1 that has more than two factors

Factor: A number that divides another number evenly with no remainder

Prime/Composite Identification Method:
  1. Count the factors: List all numbers that divide the given number evenly
  2. Compare count: If exactly 2 factors → Prime, If more than 2 → Composite
  3. Verify result: Check your factor list for completeness
Tip 1: The smallest prime number is 2 (and it's the only even prime).
Tip 2: Even numbers greater than 2 are always composite.
Tip 3: Numbers ending in 5 (except 5 itself) are composite.
Tip 4: Remember: 1 is neither prime nor composite.
Prime facts: Primes end in 1, 3, 7, or 9 (except 2 and 5). There are infinitely many primes.
Factorization: Every composite number can be broken down into prime factors uniquely.
Important Rules to Remember:

• Prime rule: Has exactly 2 factors (1 and itself)

• Composite rule: Has more than 2 factors

• Number 1: Special case - neither prime nor composite

• Even numbers: Only 2 is prime among even numbers

• Divisibility: Use divisibility rules to quickly identify factors

24 = 2³ × 3
Prime Factorization Example
Solution: Exercises 4 to 5
4 Number Classification
Exercise 4
Classify each number as prime, composite, or neither: 1, 17, 25, 29, 30, 49, 53
Definition:

Classification rules: Prime (exactly 2 factors), Composite (more than 2 factors), Neither (1 has only 1 factor)

Step 1: Analyze each number individually

1: Has only one factor (1) → Neither prime nor composite

17: Factors are 1, 17 → Prime (exactly 2 factors)

25: Factors are 1, 5, 25 → Composite (more than 2 factors)

29: Factors are 1, 29 → Prime (exactly 2 factors)

30: Factors are 1, 2, 3, 5, 6, 10, 15, 30 → Composite (8 factors)

49: Factors are 1, 7, 49 → Composite (3 factors)

53: Factors are 1, 53 → Prime (exactly 2 factors)

Step 2: Organize by classification
Prime: 17, 29, 53
Composite: 25, 30, 49
Neither: 1
Step 3: Verify each classification

Prime numbers have exactly 2 factors, composite have more than 2, 1 has only 1

Prime: 17, 29, 53 | Composite: 25, 30, 49 | Neither: 1
Final answer:

Prime: 17, 29, 53

Composite: 25, 30, 49

Neither: 1

Applied rules:

• Prime definition: Exactly 2 factors

• Composite definition: More than 2 factors

• Special case: Number 1 is neither prime nor composite

Classification Summary
PRIME
17, 29, 53
COMPOSITE
25, 30, 49
NEITHER
1
5 Advanced Prime Factorization
Exercise 5
Find the prime factorization of 60 and 72. Then find their greatest common factor (GCF).
Definition:

Greatest Common Factor (GCF): The largest number that divides both numbers evenly

Prime factorization: Breaking a number into its prime components

Step 1: Prime factorization of 60

60 = 4 × 15 = 4 × 3 × 5 = 2 × 2 × 3 × 5 = 2² × 3 × 5

Step 2: Prime factorization of 72

72 = 8 × 9 = 2 × 4 × 9 = 2 × 2 × 2 × 3 × 3 = 2³ × 3²

Step 3: Find GCF using prime factorization

60 = 2² × 3 × 5

72 = 2³ × 3²

Take lowest power of common primes: 2² × 3 = 4 × 3 = 12

Step 4: Verify GCF

60 ÷ 12 = 5 ✓

72 ÷ 12 = 6 ✓

60 = 2² × 3 × 5 | 72 = 2³ × 3² | GCF = 12
Final answer:

Prime factorization of 60: 2² × 3 × 5

Prime factorization of 72: 2³ × 3²

Greatest Common Factor: 12

Applied rules:

• Prime factorization: Break down until all factors are prime

• GCF method: Take lowest power of common prime factors

• Verification: Check that GCF divides both numbers evenly

Prime Factorization Comparison
60 = 2² × 3 × 5
Prime factors: 2, 2, 3, 5
72 = 2³ × 3²
Prime factors: 2, 2, 2, 3, 3
GCF = 2² × 3 = 12
Prime and Composite Numbers: Laws, Methods, and Definitions
Prime: 1 and itself only
Prime Number Rule
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 distinct positive divisors

Factor: A number that divides another number evenly without remainder

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

Complete Prime/Composite Methodology:
  1. Identify the number: Note if it's 1, prime, or potentially composite
  2. Find all factors: Systematically check divisibility by numbers 1 through √n
  3. Count factors: Determine if exactly 2 (prime) or more than 2 (composite)
  4. Verify result: Double-check factor list for completeness
Tip 1: To check if a number is prime, only check divisibility up to its square root.
Tip 2: All even numbers greater than 2 are composite (divisible by 2).
Tip 3: Numbers ending in 5 (except 5) are composite (divisible by 5).
Tip 4: Use divisibility rules to quickly identify factors (2, 3, 5, etc.).
Common errors: Forgetting that 1 is neither prime nor composite, miscounting factors, stopping early in factorization.
Memory aids: "Prime" sounds like "primary" - the basic building blocks. "Composite" means made up of parts.
Essential Rules and Properties:

• Prime definition: Exactly 2 distinct positive divisors

• Composite definition: More than 2 distinct positive divisors

• Number 1: Special case - neither prime nor composite

• Fundamental theorem: Every composite number has a unique prime factorization

• Smallest prime: 2 (the only even prime number)

Prime and Composite Number Patterns
Exercise 6: Prime Number Patterns
Analyze the first 30 numbers and identify prime and composite patterns.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30

Pattern Analysis: Primes become less frequent as numbers increase. All even numbers except 2 are composite.

  • Primes in range 1-30: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 (10 total)
  • Composites in range 1-30: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30 (19 total)
  • Special case: 1 is neither prime nor composite

Questions & Answers

Question: Why is 2 considered a prime number when it's even? I thought even numbers couldn't be prime.

Answer: Great observation! The number 2 is indeed special in the world of prime numbers:

  • 2 meets the exact definition of a prime number: it has exactly two factors (1 and 2)
  • It's the only even prime number because all other even numbers have at least three factors: 1, 2, and themselves
  • For example, 4 has factors 1, 2, and 4; 6 has factors 1, 2, 3, and 6

So while it's true that most even numbers are composite, 2 is the exception. It's also the smallest prime number and the only even prime, making it unique among primes.

Think of it this way: 2 is prime because it fits the definition perfectly (exactly 2 factors), not because of its parity (evenness).

Question: Why isn't 1 considered a prime number? It seems like it should be since it's only divisible by 1 and itself.

Answer: This is a common source of confusion! The number 1 is neither prime nor composite for important mathematical reasons:

  • By definition, a prime number must have exactly two distinct positive divisors
  • However, 1 only has one divisor: itself (the number 1)
  • Prime numbers need two different factors: 1 AND the number itself

Additionally, if 1 were considered prime, it would break the Fundamental Theorem of Arithmetic, which states that every integer greater than 1 can be expressed as a unique product of primes. Including 1 as prime would make this uniqueness impossible.

So 1 is in its own category - neither prime nor composite. The smallest prime number is 2.

Question: How can I quickly tell if a large number is prime without listing all its factors?

Answer: For larger numbers, here are some efficient strategies to determine if a number might be prime:

  • Check divisibility by small primes: Test if the number is divisible by 2, 3, 5, 7, 11, etc.
  • Only check up to the square root: You only need to test divisors up to √n
  • Quick divisibility tests:
    • If even → divisible by 2 (not prime unless it's 2)
    • If digits sum to multiple of 3 → divisible by 3
    • If ends in 5 or 0 → divisible by 5

For example, to check if 97 is prime: It's not even, doesn't end in 5, digit sum is 16 (not divisible by 3), and checking division by 7 gives 13.86... (not whole). Since √97 ≈ 9.8, we only need to check up to 9, so 97 is prime.

This method is much faster than listing all factors!