Solved Exercises on Prime and Composite Numbers in Grade 4

Master prime and composite numbers: identification, classification, and factorization methods through these 3 detailed exercises.

Solution: Exercises 1 to 3
1 Prime Number Identification
Exercise 1
Is 7 a prime number? Explain why or why not.
Definition:

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

Identification method:
  1. List all factors of the number
  2. Count the total number of factors
  3. If exactly 2 factors exist, it's prime
  4. If more than 2 factors exist, it's composite
Number
7
Factors
1, 7
Count
2 factors
Step 1: List all factors of 7

We check which numbers divide 7 evenly: 1 × 7 = 7

Step 2: Count the factors

Factors of 7 are: 1 and 7 (only 2 factors)

Step 3: Apply the definition

Since 7 has exactly 2 factors (1 and 7), it's a prime number

7 is a PRIME number
Final answer:

Yes, 7 is a prime number because its only factors are 1 and 7.

Applied rules:

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

• Factor finding: Check all numbers from 1 to the number itself

• Counting: Only count actual divisors

2 Composite Number Identification
Exercise 2
Is 12 a composite number? Explain why or why not.
Definition:

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

Number
12
Factors
1, 2, 3, 4, 6, 12
Count
6 factors
Step 1: List all factors of 12

We check which numbers divide 12 evenly: 1×12=12, 2×6=12, 3×4=12

Step 2: Count the factors

Factors of 12 are: 1, 2, 3, 4, 6, 12 (6 factors total)

Step 3: Apply the definition

Since 12 has more than 2 factors (6 factors), it's a composite number

12 is a COMPOSITE number
Final answer:

Yes, 12 is a composite number because it has 6 factors: 1, 2, 3, 4, 6, and 12.

Applied rules:

• Composite definition: More than 2 factors

• Factor finding: Check all numbers from 1 to the number itself

• Systematic approach: Find factor pairs efficiently

3 Special Case (Number 1)
Exercise 3
Is 1 a prime or composite number? Explain.
Definition:

Special rule: The number 1 is neither prime nor composite

Number
1
Factors
1
Classification
Neither
Step 1: List all factors of 1

Only 1 divides 1 evenly: 1 × 1 = 1

Step 2: Count the factors

Factors of 1 are: 1 (only 1 factor)

Step 3: Apply the definition

Prime numbers need exactly 2 factors, composite numbers need more than 2. Since 1 has only 1 factor, it's neither.

1 is NEITHER prime nor composite
Final answer:

The number 1 is neither prime nor composite because it has only one factor.

Applied rules:

• Special case: Number 1 has only one factor (itself)

• Prime requirement: Exactly 2 factors

• Composite requirement: More than 2 factors

Visual Representations & Examples
Prime: 2, 3, 5, 7, 11, 13...
Prime Numbers (Exactly 2 Factors)
Composite: 4, 6, 8, 9, 10, 12...
Composite Numbers (More Than 2 Factors)
2
Prime
Factors: 1, 2
3
Prime
Factors: 1, 3
4
Composite
Factors: 1, 2, 4
5
Prime
Factors: 1, 5
6
Composite
Factors: 1, 2, 3, 6
7
Prime
Factors: 1, 7
8
Composite
Factors: 1, 2, 4, 8
9
Composite
Factors: 1, 3, 9
1 is NEITHER
Special Case: Number 1
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Tip 1: The smallest prime number is 2 (and it's the only even prime).
Tip 2: All even numbers greater than 2 are composite (divisible by 2).
Tip 3: To check if a number is prime, only test divisibility up to its square root.
Rules and Methods for Prime/Composite Numbers
Prime: 2 Factors Only
Prime Numbers
Composite: >2 Factors
Composite Numbers
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 without remainder

Complete methodology:
  1. Identify the number: Determine which number you're examining
  2. Find all factors: List all numbers that divide the target number evenly
  3. Count factors: Tally the total number of factors found
  4. Classify:
    • Exactly 2 factors = Prime
    • More than 2 factors = Composite
    • Exactly 1 factor = Neither (special case: number 1)
Prime Numbers (first 20): 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71
Common Mistakes: Forgetting that 1 is neither prime nor composite, thinking all odd numbers are prime.
Important Rules to Remember:

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

• Number 1: Neither prime nor composite

• All primes >2: Are odd numbers

• Even numbers >2: Are always composite

• Quick check: Test divisibility up to √n for efficiency

Prime vs Composite Numbers Guide
2 = 1 × 2 → Prime
Prime Example
6 = 1 × 6 = 2 × 3 → Composite
Composite Example
Key definitions:

Prime: Has exactly two factors: 1 and itself

Composite: Has more than two factors

Neither: The number 1 has only one factor

Step-by-step classification:
  1. Start with 1: Check if the number is 1 (neither prime nor composite)
  2. Find factor pairs: Systematically find all numbers that multiply to give the target number
  3. Count total factors: Tally all unique factors found
  4. Compare with definition: 2 factors = prime, more than 2 = composite
Tip 1: Use divisibility rules to quickly identify factors (2, 3, 5, etc.)
Tip 2: Stop checking at the square root of the number for efficiency
Tip 3: Remember that 2 is the only even prime number
Tip 4: Practice with numbers 1-20 to build fluency
Prime Patterns: Primes become less frequent as numbers get larger, but there are infinitely many primes.
Real-world use: Prime numbers are crucial in cryptography and computer security.
Classification Chart:

• Prime (2 factors): 2, 3, 5, 7, 11, 13, 17, 19, 23, 29...

• Composite (>2 factors): 4, 6, 8, 9, 10, 12, 14, 15, 16, 18...

• Neither (1 factor): 1

Questions & Answers

Question: Why is 2 considered a prime number when it's an even number? I thought prime numbers were odd.

Answer: Great observation! 2 is indeed the only even prime number, and here's why:

  • 2 has exactly two factors: 1 and 2 (which meets the prime definition)
  • Every other even number has at least three factors: 1, 2, and itself
  • For example, 4 has factors: 1, 2, and 4 (so it's composite)

So while it's true that most prime numbers are odd, 2 is the exception because it's divisible only by 1 and itself. It's also the smallest prime number!

Rule: 2 is the only even prime number. All other even numbers greater than 2 are composite.

Question: Is 9 a prime number? I thought odd numbers were usually prime, but I heard 9 might not be.

Answer: Good question! While many small odd numbers are prime, not all odd numbers are prime. Let's check 9:

  • Factors of 9: 1, 3, and 9
  • That's 3 factors total (more than 2)
  • Since prime numbers must have exactly 2 factors, 9 is composite

To find factors of 9: 1×9=9 and 3×3=9, so the factors are 1, 3, and 9.

Common mistake: Thinking that all odd numbers are prime. Remember to always check the number of factors!

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

Answer: Here are some quick strategies for checking if a number is prime:

  • Even numbers > 2: Always composite (divisible by 2)
  • Ends in 5 (except 5): Composite (divisible by 5)
  • Sum of digits divisible by 3: Composite (divisible by 3)
  • Check divisibility: Only test division by primes up to the square root of the number

Example: To check if 29 is prime, only test division by primes ≤ √29 (about 5.4), so test 2, 3, and 5: 29÷2 = 14.5, 29÷3 = 9.67, 29÷5 = 5.8. Since none divide evenly, 29 is prime!

For 4th grade, focus on memorizing primes up to 20: 2, 3, 5, 7, 11, 13, 17, 19.