Solved Exercises on Prime and Composite Numbers in Grade 4

Master prime and composite numbers: factors, divisibility, patterns and number properties 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, 3, 4, 5, 6, 7, 8, 9, 10, 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

Method to identify primes:
  1. List all factors of each number
  2. Count the number of factors
  3. If exactly 2 factors → Prime
  4. If more than 2 factors → Composite
Numbers
2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13
Analysis
Prime or Composite
Step 1: Analyze each number's factors

2: Factors = {1, 2} → Exactly 2 factors → Prime ✅

3: Factors = {1, 3} → Exactly 2 factors → Prime ✅

4: Factors = {1, 2, 4} → 3 factors → Composite ❌

5: Factors = {1, 5} → Exactly 2 factors → Prime ✅

6: Factors = {1, 2, 3, 6} → 4 factors → Composite ❌

7: Factors = {1, 7} → Exactly 2 factors → Prime ✅

8: Factors = {1, 2, 4, 8} → 4 factors → Composite ❌

9: Factors = {1, 3, 9} → 3 factors → Composite ❌

10: Factors = {1, 2, 5, 10} → 4 factors → Composite ❌

11: Factors = {1, 11} → Exactly 2 factors → Prime ✅

12: Factors = {1, 2, 3, 4, 6, 12} → 6 factors → Composite ❌

13: Factors = {1, 13} → Exactly 2 factors → Prime ✅

Step 2: List all prime numbers found

Prime numbers: 2, 3, 5, 7, 11, 13

Step 3: Remember key facts

• 2 is the smallest prime number

• 2 is the only even prime number

• 1 is neither prime nor composite

Prime numbers: 2, 3, 5, 7, 11, 13
Final answer:

Prime numbers: 2, 3, 5, 7, 11, 13

Composite numbers: 4, 6, 8, 9, 10, 12

Applied rules:

• Prime Rule: Has exactly two factors: 1 and itself

• Composite Rule: Has more than two factors

• Special Case: 1 is neither prime nor composite

2 Finding Factors
Exercise 2
Find all factors of 24. Then classify 24 as prime or composite.
Definition:

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

Factor Pairs: Two numbers that multiply together to make the target number

Number
24
Factors
1, 2, 3, 4, 6, 8, 12, 24
Classification
Composite
Step 1: Find factor pairs systematically

Start with 1: 1 × 24 = 24 → Factors: 1, 24

Try 2: 2 × 12 = 24 → Factors: 2, 12

Try 3: 3 × 8 = 24 → Factors: 3, 8

Try 4: 4 × 6 = 24 → Factors: 4, 6

Try 5: 5 doesn't divide 24 evenly

Stop when factors repeat (we already have 6)

Step 2: List all factors in order

All factors of 24: {1, 2, 3, 4, 6, 8, 12, 24}

Step 3: Count factors and classify

Number of factors: 8 (more than 2)

Since 24 has more than 2 factors → 24 is composite

Factors of 24: {1, 2, 3, 4, 6, 8, 12, 24}
Classification: Composite
Final answer:

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

24 is a composite number because it has 8 factors (more than 2)

Applied rules:

• Systematic Method: Check divisors from 1 upward

• Factor Pair Strategy: Find two numbers that multiply to target

• Stopping Point: Stop when factors start repeating

3 Prime vs Composite Identification
Exercise 3
Classify each number as prime or composite:
17, 21, 29, 33, 37
Definition:

Quick Prime Test: Try dividing by 2, 3, 5, 7, 11, etc. up to √n

Numbers
17, 21, 29, 33, 37
Classification
Mixed
Step 1: Test 17

Check divisibility by small primes: 2, 3, 5

17 ÷ 2 = 8.5 (not whole)

17 ÷ 3 = 5.67 (not whole)

17 ÷ 5 = 3.4 (not whole)

No divisors found → 17 is prime

Step 2: Test 21

21 ÷ 3 = 7 (whole number)

So 21 = 3 × 7 → 21 has factors 1, 3, 7, 21

More than 2 factors → 21 is composite

Step 3: Test 29

Check divisibility by 2, 3, 5

29 ÷ 2 = 14.5 (not whole)

29 ÷ 3 = 9.67 (not whole)

29 ÷ 5 = 5.8 (not whole)

No divisors found → 29 is prime

Step 4: Test 33

33 ÷ 3 = 11 (whole number)

So 33 = 3 × 11 → 33 has factors 1, 3, 11, 33

More than 2 factors → 33 is composite

Step 5: Test 37

Check divisibility by 2, 3, 5

37 ÷ 2 = 18.5 (not whole)

37 ÷ 3 = 12.33 (not whole)

37 ÷ 5 = 7.4 (not whole)

No divisors found → 37 is prime

Prime: 17, 29, 37
Composite: 21, 33
Final answer:

Prime numbers: 17, 29, 37

Composite numbers: 21, 33

Applied rules:

• Prime Testing: Check divisibility by small primes up to √n

• Even Numbers: All even numbers > 2 are composite

• Divisibility Rules: Use quick checks for 2, 3, 5

Prime and Composite Numbers Guide
Prime: Exactly 2 factors
Definition
Prime Numbers
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47
First 15 prime numbers
Composite Numbers
4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25
First 15 composite numbers
Special Cases
1 = Neither Prime Nor Composite
Unique classification
Key definitions:

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

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

Complete methodology:
  1. Identify the number: Look at the target number to test
  2. Find all factors: List factor pairs systematically
  3. Count factors: Determine how many factors exist
  4. Classify: If exactly 2 factors → Prime, if more than 2 → Composite
Tip 1: 2 is the only even prime number. All other even numbers are composite.
Tip 2: Numbers ending in 5 (except 5 itself) are always composite.
Tip 3: To test if large numbers are prime, only check divisibility up to √n.
Tip 4: 1 is neither prime nor composite - it's a special case!
Common errors: Forgetting that 1 is neither prime nor composite, thinking even numbers can be prime (except 2).
Memory aid: "Prime" sounds like "primary" - the basic building blocks of numbers.
Essential facts to remember:

• Smallest prime number: 2

• Only even prime: 2

• 1 is neither prime nor composite

• Prime numbers are the "building blocks" of all numbers

• Every composite number can be made by multiplying prime numbers

Solution: Exercises 4 to 5
4 Pattern Recognition
Exercise 4
Look at the pattern: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ...
What type of numbers are these? What comes next?
Definition:

Prime Number Sequence: The ordered list of all prime numbers

Pattern
2, 3, 5, 7, 11, 13, 17, 19, 23, 29
Type
Prime Numbers
Next
31, 37
Step 1: Identify the pattern

2, 3, 5, 7, 11, 13, 17, 19, 23, 29

These are all prime numbers in ascending order!

Step 2: Verify each number is prime

2: Factors {1, 2} → Prime

3: Factors {1, 3} → Prime

5: Factors {1, 5} → Prime

7: Factors {1, 7} → Prime

11: Factors {1, 11} → Prime

13: Factors {1, 13} → Prime

17: Factors {1, 17} → Prime

19: Factors {1, 19} → Prime

23: Factors {1, 23} → Prime

29: Factors {1, 29} → Prime

Step 3: Find the next prime numbers

After 29, check: 30 (composite, 2×15), 31 (try dividing by 2, 3, 5)

31 ÷ 2 = 15.5, 31 ÷ 3 = 10.33, 31 ÷ 5 = 6.2 → 31 is prime

After 31: 32 (even), 33 (3×11), 34 (even), 35 (5×7), 36 (even), 37

37 ÷ 2 = 18.5, 37 ÷ 3 = 12.33, 37 ÷ 5 = 7.4 → 37 is prime

Pattern: Prime numbers in order
Next numbers: 31, 37, 41, 43, 47...
Final answer:

This is the sequence of prime numbers in ascending order.

The next numbers in the sequence are: 31, 37, 41, 43, 47...

Applied rules:

• Pattern Recognition: Look for mathematical relationships

• Prime Testing: Verify each number meets prime criteria

• Sequence Continuation: Apply same rule to find next terms

5 Problem Solving
Exercise 5
Sarah has 12 stickers. She wants to arrange them in equal rows. How many different ways can she arrange them? Is 12 prime or composite?
Definition:

Equal Arrangement: Using factors to determine possible rectangular arrangements

Stickers
12
Factor Pairs
1×12, 2×6, 3×4
Classification
Composite
Step 1: Find all factor pairs of 12

1 × 12 = 12 → 1 row of 12, or 12 rows of 1

2 × 6 = 12 → 2 rows of 6, or 6 rows of 2

3 × 4 = 12 → 3 rows of 4, or 4 rows of 3

Step 2: List all possible arrangements

• 1 row × 12 columns

• 2 rows × 6 columns

• 3 rows × 4 columns

• 4 rows × 3 columns

• 6 rows × 2 columns

• 12 rows × 1 column

Total: 6 different ways

Step 3: Count total factors and classify

Factors of 12: {1, 2, 3, 4, 6, 12}

Number of factors: 6 (more than 2)

Since 12 has more than 2 factors → 12 is composite

Step 4: Connect to real-world context

The number of arrangements equals the number of factor pairs

More factors = more arrangement possibilities

Ways to arrange: 6
Classification: Composite
Factors: {1, 2, 3, 4, 6, 12}
Final answer:

Sarah can arrange her 12 stickers in 6 different ways:

1×12, 2×6, 3×4, 4×3, 6×2, 12×1

12 is a composite number because it has 6 factors (more than 2).

Applied rules:

• Real-world Application: Factors relate to arranging objects

• Factor Pairs: Each pair represents a possible arrangement

• Classification: More than 2 factors = composite

Prime and Composite Number Properties
Prime: 2 factors only
Core Property
Prime Patterns
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47
Primes up to 50
Composite Patterns
4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25
Composites up to 25
Prime Testing
Check divisors up to √n
Efficient method
Key definitions:

Prime Factorization: Expressing a composite number as a product of primes

Factor Tree: A visual way to find prime factors

Divisibility Rules: Quick tests for division without calculation

Prime testing strategies:
  1. Even Check: If number > 2 and even → composite
  2. Digital Root: If sum of digits divisible by 3 → divisible by 3
  3. Ends in 5: If ends in 5 (and > 5) → divisible by 5
  4. Square Root Test: Only check divisors up to √n
Tip 1: Use divisibility rules to quickly eliminate composites.
Tip 2: Twin primes are primes that differ by 2 (like 11 and 13).
Tip 3: The number of primes decreases as numbers get larger.
Tip 4: Every number can be expressed as a unique product of primes.
Interesting Fact: There are infinitely many prime numbers - proven by Euclid!
Real-world Use: Primes are essential for computer security and encryption.
Important rules for memorization:

• All primes > 3 are of the form 6n±1

• The sum of digits test: if sum divisible by 3, number divisible by 3

• If last digit is 0 or 5, number divisible by 5

• Prime factorization is unique for every number

• There are 25 primes less than 100

Questions & Answers

Question: Why is 1 not considered a prime number? It only has one factor which is itself.

Answer: Great question! The definition of a prime number specifically states it must have exactly TWO factors: 1 and itself. Since 1 only has ONE factor (which is 1), it doesn't meet the definition of prime.

Additionally, if 1 were considered prime, it would break important mathematical rules. For example, the Fundamental Theorem of Arithmetic says every number can be written as a unique product of primes. If 1 were prime, we could write 6 as 2×3 or 1×2×3 or 1×1×2×3, etc., breaking uniqueness.

So 1 gets its own special category: neither prime nor composite!

Question: How can I quickly tell if a big number like 47 is prime without checking every number?

Answer: You only need to check divisibility by prime numbers up to the square root of your number! For 47:

  • √47 ≈ 6.86, so only check primes ≤ 6: that's just 2, 3, and 5
  • 47 ÷ 2 = 23.5 (not whole)
  • 47 ÷ 3 = 15.67 (not whole)
  • 47 ÷ 5 = 9.4 (not whole)

Since none divide evenly, 47 is prime! This saves tons of time compared to checking every number.

Question: Why are prime numbers important? When will I ever use them?

Answer: Prime numbers are incredibly important! Here are some uses:

  • Cryptography: Computer security and passwords rely on large prime numbers
  • Number Theory: Primes are the "building blocks" of all numbers
  • Problem Solving: Understanding primes helps with factoring, which is needed for fractions, ratios, and advanced math
  • Patterns: Studying primes helps mathematicians understand how numbers work

Think of primes as the "DNA" of numbers - understanding them helps unlock more complex mathematics!