Solved Exercises on Prime and Composite Numbers in Grade 4

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

Prime and Composite Concepts
Prime and Composite Numbers Explained
Prime numbers have exactly two factors: 1 and themselves. Composite numbers have more than two factors. Understanding these classifications helps with factoring, multiples, and recognizing number patterns. The number 1 is neither prime nor composite.
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Prime: Only 1 and itself
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Composite: Many factors
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Special: 1 (neither)
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Method: Factor count
2, 3, 5, 7, 11... 👈 Prime | Composite 👉 4, 6, 8, 9, 10...
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

Special Case: The number 1 is neither prime nor composite

Identification Method:
  1. List all factors: Find all numbers that divide the target evenly
  2. Count factors: Count how many factors the number has
  3. Classify: If exactly 2 factors → Prime; if more than 2 → Composite
Tip 1: Even numbers (except 2) are always composite.
Tip 2: Any number ending in 5 (except 5) is composite.
Tip 3: Start checking divisibility by small primes (2, 3, 5, 7...).
Important: The first 10 prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29
Exercise Solutions 1-2
1 Identifying Prime Numbers
Exercise 1
Is 17 a prime or composite number? Explain your reasoning.
Problem Type:

Prime identification: Finding all factors of a number

Solution Strategy:
  1. Find all factors of 17 by checking divisibility
  2. Count the total number of factors
  3. Determine if it's prime (exactly 2) or composite (more than 2)
Step 1: Check divisibility by numbers 1 to √17

√17 ≈ 4.12, so check: 1, 2, 3, 4

17 ÷ 1 = 17 (remainder 0) → Factor: 1, 17

17 ÷ 2 = 8 remainder 1 → Not a factor

17 ÷ 3 = 5 remainder 2 → Not a factor

17 ÷ 4 = 4 remainder 1 → Not a factor

Step 2: List all factors

Factors of 17: 1, 17

Step 3: Count factors

There are exactly 2 factors: 1 and 17

17 is a prime number.
Final Answer:

17 is a prime number because it has exactly two factors: 1 and 17.

Applied Rules:

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

• Efficiency rule: Only check up to √n for factors

• Factor pairs: If a is a factor, then n/a is also a factor

2 Identifying Composite Numbers
Exercise 2
Is 24 a prime or composite number? Find all its factors.
Problem Type:

Composite identification: Finding multiple factors of a number

Step 1: Find all factors systematically

Check divisibility by numbers 1 to √24

√24 ≈ 4.9, so check: 1, 2, 3, 4

24 ÷ 1 = 24 → Factors: 1, 24

24 ÷ 2 = 12 → Factors: 2, 12

24 ÷ 3 = 8 → Factors: 3, 8

24 ÷ 4 = 6 → Factors: 4, 6

Step 2: List all unique factors

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

Step 3: Count factors and classify

There are 8 factors, which is more than 2

24 is a composite number.
Final Answer:

24 is a composite number because it has more than two factors: 1, 2, 3, 4, 6, 8, 12, 24.

Applied Rules:

• Composite definition: More than 2 factors

• Factor pairs: Each factor has a corresponding pair

• Systematic approach: Check only up to √n for efficiency

Exercise 3 & Practice Problems
3 Special Cases and Patterns
Exercise 3
Classify these numbers: 1, 2, 9, 15. Explain why each number is classified as it is.
Problem Type:

Multiple classifications: Including special cases and patterns

Number 1: Special case

Factors of 1: Only 1

Since it has only 1 factor (not exactly 2), it's neither prime nor composite.

Number 2: Smallest prime

Factors of 2: 1, 2

Exactly 2 factors → Prime number (also the only even prime!)

Number 9: Odd composite

Factors of 9: 1, 3, 9

More than 2 factors → Composite number

Number 15: Product of primes

Factors of 15: 1, 3, 5, 15

More than 2 factors → Composite number

1: Neither | 2: Prime | 9: Composite | 15: Composite
Final Answer:

1: Neither prime nor composite (has only 1 factor). 2: Prime (has exactly 2 factors). 9: Composite (has 3 factors). 15: Composite (has 4 factors).

Applied Rules:

• Special case: 1 has only one factor, so it's neither

• Even primes: 2 is the only even prime number

• Odd composites: Many odd numbers are composite

Key Notes
  • Prime numbers have exactly 2 factors: 1 and the number itself
  • Composite numbers have 3 or more factors
  • The number 1 is neither prime nor composite
  • 2 is the only even prime number
Common Mistakes
  • Thinking 1 is a prime number (it's neither)
  • Believing all odd numbers are prime
  • Missing factor pairs when listing factors
  • Not stopping at √n when checking for factors
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Prime: Factors = {1, n} | Composite: Factors > 2
Classification Formula
Complete Summary: Prime and Composite Numbers

Definitions of Key Concepts:

  • Prime Number: A whole number greater than 1 that has exactly two distinct positive factors: 1 and itself
  • Composite Number: A whole number greater than 1 that has more than two distinct positive factors
  • Factor: A number that divides another number evenly without leaving a remainder
  • Neither Prime nor Composite: The number 1, which has only one factor
  • Factor Pair: Two numbers that multiply together to give the original number

Core Rules and Principles:

  • Prime Rule: Exactly 2 factors (1 and the number itself)
  • Composite Rule: More than 2 factors
  • Special Case Rule: 1 has only 1 factor, so it's neither prime nor composite
  • Even Number Rule: All even numbers except 2 are composite
  • Square Root Rule: Only check for factors up to √n for efficiency

Step-by-Step Methods:

  1. Systematic Factor Finding: Check divisibility starting from 1 up to √n
  2. Record Factor Pairs: For each factor found, record both the divisor and quotient
  3. Count Total Factors: Tally all unique factors found
  4. Classify the Number: If exactly 2 factors → Prime; if more than 2 → Composite
  5. Verify Results: Double-check by ensuring all factors divide the number evenly

Examples - Simple to Advanced:

  • Simple Prime: 5 (factors: 1, 5) → Prime
  • Simple Composite: 6 (factors: 1, 2, 3, 6) → Composite
  • Intermediate Prime: 13 (factors: 1, 13) → Prime
  • Intermediate Composite: 18 (factors: 1, 2, 3, 6, 9, 18) → Composite
  • Advanced Composite: 36 (factors: 1, 2, 3, 4, 6, 9, 12, 18, 36) → Composite

Tips and Tricks:

  • Quick Check for Even Numbers: All even numbers > 2 are composite (divisible by 2)
  • End in 5 Rule: Numbers ending in 5 (except 5 itself) are composite
  • Sum of Digits Rule: If sum of digits is divisible by 3, the number is divisible by 3
  • Square Root Efficiency: Only check factors up to √n to save time
  • Memorize Small Primes: Know the first 10 primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29

Common Pitfalls:

  • Mistaking 1 for a prime number (it has only 1 factor, not 2)
  • Assuming all odd numbers are prime (e.g., 9, 15, 21 are composite)
  • Missing factor pairs when listing factors systematically
  • Not checking divisibility by all relevant numbers up to √n
  • Confusing factors with multiples

Key Takeaways for Memorization:

  • Remember: Prime = 2 factors only (1 and itself)
  • Remember: Composite = More than 2 factors
  • Remember: 1 is special = Neither prime nor composite
  • Remember: 2 is the only even prime
  • Remember: Check up to √n for efficiency

Questions & Answers

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

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

  • Prime definition: A prime number has exactly two factors: 1 and itself
  • For 2: The factors are 1 and 2 (exactly 2 factors) → Prime
  • Why it's special: 2 is divisible by 1 and 2 only, so it meets the prime criteria
  • All other even numbers: Have at least three factors: 1, 2, and themselves → Composite

For example: 4 has factors 1, 2, 4 (3 factors → Composite), 6 has factors 1, 2, 3, 6 (4 factors → Composite).

So remember: 2 is the only even prime number. All other even numbers greater than 2 are composite because they're divisible by 2.

Question: My child is having trouble finding all the factors efficiently. What's the best way to organize this process?

Answer: Here's an efficient method to find all factors systematically:

  1. Start with 1: Every number has 1 and itself as factors
  2. Check sequentially: Test divisibility by 2, 3, 4, etc., up to √n
  3. Record pairs: For each factor found, record both the divisor and quotient
  4. Stop at √n: Once you pass √n, you'll just find the reverse of pairs already recorded

For example, finding factors of 24:

  • 1: 1 × 24 → Record 1, 24
  • 2: 2 × 12 → Record 2, 12
  • 3: 3 × 8 → Record 3, 8
  • 4: 4 × 6 → Record 4, 6
  • √24 ≈ 4.9, so stop here
  • All factors: 1, 2, 3, 4, 6, 8, 12, 24

This method ensures you don't miss any factors and avoids redundant work.

Question: How can I help students remember the first 10 prime numbers easily?

Answer: Here are effective memory strategies for the first 10 prime numbers (2, 3, 5, 7, 11, 13, 17, 19, 23, 29):

  • Pattern Recognition: Notice that after 2, all primes are odd numbers
  • Rhyme Method: "Two, three, five, seven, eleven primes are heaven!"
  • Grouping Strategy: Remember them in chunks: (2, 3, 5, 7) and (11, 13, 17, 19, 23, 29)
  • Visual Memory: Create a prime number chart and practice skip counting
  • Practice Games: Use flashcards or prime number identification games

Also emphasize that students should especially remember: 2 is the smallest prime, and it's the only even prime number. This helps distinguish it from other primes.

Regular practice with identifying primes in various ranges will strengthen their recall over time.