Solved Exercises on Prime and Composite Numbers Examples in Grade 4

Master prime and composite numbers through examples: identifying primes, finding composites, understanding divisibility, and recognizing number properties through these 5 detailed exercises.

Example Solutions: 1 to 3
1 Example: Prime Number
Example 1
Is 11 a prime number? Show all factors and explain why.
Definition:

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

Method:
  1. Start with 1 and check if it divides the number evenly
  2. Check 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 (up to the number itself)
  3. List all numbers that divide evenly
  4. Count the total factors
  5. If exactly 2 factors exist, it's prime
Number
11
Factors
1, 11
Count
2 factors
Step 1: Check divisibility by 1

11 ÷ 1 = 11 remainder 0 → 1 is a factor

Step 2: Check divisibility by 2

11 ÷ 2 = 5 remainder 1 → 2 is not a factor

Step 3: Check divisibility by 3

11 ÷ 3 = 3 remainder 2 → 3 is not a factor

Step 4: Check divisibility by 4

11 ÷ 4 = 2 remainder 3 → 4 is not a factor

Step 5: Check divisibility by 5

11 ÷ 5 = 2 remainder 1 → 5 is not a factor

Step 6: Check divisibility by 6

11 ÷ 6 = 1 remainder 5 → 6 is not a factor

Step 7: Check divisibility by 7

11 ÷ 7 = 1 remainder 4 → 7 is not a factor

Step 8: Check divisibility by 8

11 ÷ 8 = 1 remainder 3 → 8 is not a factor

Step 9: Check divisibility by 9

11 ÷ 9 = 1 remainder 2 → 9 is not a factor

Step 10: Check divisibility by 10

11 ÷ 10 = 1 remainder 1 → 10 is not a factor

Step 11: Check divisibility by 11

11 ÷ 11 = 1 remainder 0 → 11 is a factor

Factors of 11: 1, 11 (exactly 2 factors)
Final answer:

Yes, 11 is a prime number because it has exactly two factors: 1 and 11.

Applied rules:

• Prime Definition: Must have exactly two factors

• Factor Testing: Check divisibility systematically

• Counting: Only 2 factors means it's prime

2 Example: Composite Number
Example 2
Is 12 a composite number? Find all its factors and explain why.
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: Check divisibility by 1

12 ÷ 1 = 12 remainder 0 → 1 is a factor

Step 2: Check divisibility by 2

12 ÷ 2 = 6 remainder 0 → 2 is a factor

Step 3: Check divisibility by 3

12 ÷ 3 = 4 remainder 0 → 3 is a factor

Step 4: Check divisibility by 4

12 ÷ 4 = 3 remainder 0 → 4 is a factor

Step 5: Check divisibility by 5

12 ÷ 5 = 2 remainder 2 → 5 is not a factor

Step 6: Check divisibility by 6

12 ÷ 6 = 2 remainder 0 → 6 is a factor

Step 7: Check divisibility by 7

12 ÷ 7 = 1 remainder 5 → 7 is not a factor

Step 8: Check divisibility by 8

12 ÷ 8 = 1 remainder 4 → 8 is not a factor

Step 9: Check divisibility by 9

12 ÷ 9 = 1 remainder 3 → 9 is not a factor

Step 10: Check divisibility by 10

12 ÷ 10 = 1 remainder 2 → 10 is not a factor

Step 11: Check divisibility by 11

12 ÷ 11 = 1 remainder 1 → 11 is not a factor

Step 12: Check divisibility by 12

12 ÷ 12 = 1 remainder 0 → 12 is a factor

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

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

Applied rules:

• Composite Definition: Must have more than two factors

• Factor Pairs: 1×12=12, 2×6=12, 3×4=12

• Counting: More than 2 factors means it's composite

3 Example: Classification
Example 3
From the list below, classify each number as prime or composite: 4, 7, 8, 13, 15
Definition:

Classification: Determining if each number is prime or composite by finding its factors

Numbers
4, 7, 8, 13, 15
Factors
List each
Types
C, P, C, P, C
Step 1: Analyze 4

Factors: 1, 2, 4 → Composite (3 factors)

Step 2: Analyze 7

Factors: 1, 7 → Prime (exactly 2 factors)

Step 3: Analyze 8

Factors: 1, 2, 4, 8 → Composite (4 factors)

Step 4: Analyze 13

Factors: 1, 13 → Prime (exactly 2 factors)

Step 5: Analyze 15

Factors: 1, 3, 5, 15 → Composite (4 factors)

Step 6: Final classification

Prime: 7, 13 | Composite: 4, 8, 15

Prime: 7, 13 | Composite: 4, 8, 15
Final answer:

Prime numbers: 7, 13 | Composite numbers: 4, 8, 15

Applied rules:

• Systematic Analysis: Examine each number individually

• Factor Counting: Determine the exact number of factors

• Classification Rules: 2 factors = prime, more than 2 = composite

Examples & Methods
Prime: 2 Factors | Composite: >2 Factors
Classification Rule
Example 1
Prime: 5 = factors 1, 5
Prime Example
Example 2
Composite: 6 = factors 1, 2, 3, 6
Composite Example
Example 3
Special: 1 = only factor 1
Special Case
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

Factors: Numbers that divide evenly into another number without leaving a remainder

Methodology:
  1. Check if number > 1: Prime/composite only applies to numbers greater than 1
  2. Find all factors: Systematically check divisibility from 1 up to the number
  3. Count factors: Tally the total number of factors found
  4. Classify: 2 factors = prime, more than 2 = composite
Tip 1: Even numbers (except 2) are always composite since they're divisible by 2.
Tip 2: Numbers ending in 5 (except 5) are composite since they're divisible by 5.
Tip 3: The number 2 is the only even prime number.
Tip 4: To save time, only test divisors up to the square root of the number.
Common errors: Thinking 1 is prime, missing factors when counting, confusing factors with multiples.
Key concepts: Prime numbers are the building blocks of all numbers, every composite number can be broken down into primes.
Rules to remember:

• 1 is neither prime nor composite

• 2 is the smallest and only even prime number

• All primes greater than 2 are odd numbers

• The first few prime numbers: 2, 3, 5, 7, 11, 13, 17, 19...

• Every composite number has at least one prime factor

Example Solutions: 4 to 5
4 Example: Word Problem
Example 4
Sarah has 17 cookies. She wants to share them equally among some friends with no cookies left over. How many friends can she share with? Is 17 prime or composite?
Definition:

Word problems: Real-world situations requiring knowledge of factors and divisibility

Cookie count
17
Factors
1, 17
Sharing options
1 friend or 17 friends
Step 1: Find factors of 17

Factors of 17: 1, 17

Step 2: Interpret sharing options

Option 1: Share with 1 friend (17 ÷ 1 = 17 cookies each)

Option 2: Share with 17 friends (17 ÷ 17 = 1 cookie each)

Step 3: Determine if 17 is prime or composite

Since 17 has exactly 2 factors (1 and 17), it's prime

Step 4: Answer the question

Sarah can share with 1 friend (giving 17 cookies) or 17 friends (giving 1 cookie each)

17 is prime, so Sarah can share with 1 or 17 friends
Final answer:

Sarah can share her 17 cookies with 1 friend (17 cookies each) or 17 friends (1 cookie each). The number 17 is prime because it has exactly two factors: 1 and 17.

Applied rules:

• Prime Application: Prime numbers have limited sharing options

• Real-world connection: Equal sharing requires understanding factors

• Verification: Confirm that sharing uses all items

5 Example: Pattern Recognition
Example 5
Find all prime numbers between 20 and 30. List them and explain your reasoning.
Definition:

Range analysis: Systematically checking numbers within a specified range

Range
20-30
Check each
21, 22, 23, 24, 25, 26, 27, 28, 29
Primes
23, 29
Step 1: List numbers 21-29 (excluding 20 and 30)

21, 22, 23, 24, 25, 26, 27, 28, 29

Step 2: Check 21

21 = 3 × 7 → composite

Step 3: Check 22

22 = 2 × 11 → composite

Step 4: Check 23

Factors: 1, 23 → prime

Step 5: Check 24

24 = 2 × 12 → composite

Step 6: Check 25

25 = 5 × 5 → composite

Step 7: Check 26

26 = 2 × 13 → composite

Step 8: Check 27

27 = 3 × 9 → composite

Step 9: Check 28

28 = 4 × 7 → composite

Step 10: Check 29

Factors: 1, 29 → prime

Prime numbers between 20 and 30: 23, 29
Final answer:

The prime numbers between 20 and 30 are: 23 and 29

Applied rules:

• Systematic approach: Check each number in the range individually

• Even number rule: All even numbers > 2 are composite

• Verification: Double-check each classification

Examples and Key Concepts
Prime: 2 Factors | Composite: >2 Factors
Classification Examples
Example definitions:

Prime Number Examples: 2 (factors: 1, 2), 3 (factors: 1, 3), 5 (factors: 1, 5), 7 (factors: 1, 7), 11 (factors: 1, 11), 13 (factors: 1, 13)

Composite Number Examples: 4 (factors: 1, 2, 4), 6 (factors: 1, 2, 3, 6), 8 (factors: 1, 2, 4, 8), 9 (factors: 1, 3, 9), 10 (factors: 1, 2, 5, 10)

Special Cases: 1 has only one factor (itself), so it's neither prime nor composite

Example approach:
  1. Example 1 (Prime): Take number 7 → factors are 1 and 7 → exactly 2 factors → prime
  2. Example 2 (Composite): Take number 8 → factors are 1, 2, 4, 8 → 4 factors → composite
  3. Example 3 (Pattern): Notice that all even numbers > 2 are composite
  4. Example 4 (Application): Prime numbers limit how items can be shared equally
Example Tip 1: The first few primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29...
Example Tip 2: All primes > 2 are odd numbers (since even numbers > 2 are divisible by 2)
Example Tip 3: Primes become less frequent as numbers get larger
Example Tip 4: Every composite number can be broken down into prime factors
Example Properties: Prime numbers are like building blocks - everything else is made from them. Composite numbers are like "group projects" made up of smaller prime numbers.
Real-world Example: Computer security uses large prime numbers to keep passwords and messages safe.
Example rules:

• 2 is the only even prime number

• All primes > 2 are odd numbers

• Numbers ending in 5 (except 5) are composite

• Every number > 1 is either prime or composite

• Prime factorization is unique for each number

Questions & Answers

Question: Why is 1 not a prime number? It seems like it should be since it only divides by itself.

Answer: This is a great question that confuses many students!

The definition of a prime number specifically states it must have exactly two different factors: 1 and itself. Since 1 only has one factor (itself), it doesn't meet this requirement.

Think of it like this: prime numbers are special because they can only be arranged in exactly two rectangular ways (1 row or 1 column). The number 1 can only be arranged in one way, so it gets its own special category.

So 1 is neither prime nor composite - it's just special by itself!

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

Answer: You can use several smart strategies to make this easier!

  • Check divisibility by small primes first: 2, 3, 5, 7, 11
  • Only check up to the square root: If n is your number, only test divisors up to √n
  • Use divisibility rules: If the sum of digits is divisible by 3, the number is divisible by 3

For example, to check if 47 is prime:

  • It's not even, so not divisible by 2
  • Sum of digits: 4+7=11, not divisible by 3
  • Doesn't end in 0 or 5, so not divisible by 5
  • √47 ≈ 6.9, so we only check up to 6
  • Check 7: 47 ÷ 7 = 6 remainder 5 → not divisible

Since we've checked all primes up to √47 and none divide 47, it's prime!

Question: Are there any patterns to help me remember which numbers are prime?

Answer: Yes, there are helpful patterns to recognize prime numbers!

  • All primes > 2 are odd (except 2, which is the only even prime)
  • Primes > 5 end in 1, 3, 7, or 9 (never 0, 2, 4, 5, 6, 8)
  • Primes become less frequent as numbers get larger

Here are the first 15 primes to help you memorize: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47

Notice that after 2 and 5, all primes end in 1, 3, 7, or 9!