Solved Exercises on Prime and Composite Numbers Exercises in Grade 4

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

Exercise Solutions: 1 to 3
1 Exercise: Prime Identification
Exercise 1
Is 19 a prime number? Show all factors and explain your reasoning.
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, 12, 13, 14, 15, 16, 17, 18, 19 (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
19
Factors
1, 19
Count
2 factors
Step 1: Check divisibility by 1

19 ÷ 1 = 19 remainder 0 → 1 is a factor

Step 2: Check divisibility by 2

19 ÷ 2 = 9 remainder 1 → 2 is not a factor

Step 3: Check divisibility by 3

19 ÷ 3 = 6 remainder 1 → 3 is not a factor

Step 4: Check divisibility by 4

19 ÷ 4 = 4 remainder 3 → 4 is not a factor

Step 5: Check divisibility by 5

19 ÷ 5 = 3 remainder 4 → 5 is not a factor

Step 6: Check divisibility by 6

19 ÷ 6 = 3 remainder 1 → 6 is not a factor

Step 7: Check divisibility by 7

19 ÷ 7 = 2 remainder 5 → 7 is not a factor

Step 8: Check divisibility by 8

19 ÷ 8 = 2 remainder 3 → 8 is not a factor

Step 9: Check divisibility by 9

19 ÷ 9 = 2 remainder 1 → 9 is not a factor

Step 10: Check divisibility by 10

19 ÷ 10 = 1 remainder 9 → 10 is not a factor

Step 11: Check divisibility by 11

19 ÷ 11 = 1 remainder 8 → 11 is not a factor

Step 12: Check divisibility by 12

19 ÷ 12 = 1 remainder 7 → 12 is not a factor

Step 13: Check divisibility by 13

19 ÷ 13 = 1 remainder 6 → 13 is not a factor

Step 14: Check divisibility by 14

19 ÷ 14 = 1 remainder 5 → 14 is not a factor

Step 15: Check divisibility by 15

19 ÷ 15 = 1 remainder 4 → 15 is not a factor

Step 16: Check divisibility by 16

19 ÷ 16 = 1 remainder 3 → 16 is not a factor

Step 17: Check divisibility by 17

19 ÷ 17 = 1 remainder 2 → 17 is not a factor

Step 18: Check divisibility by 18

19 ÷ 18 = 1 remainder 1 → 18 is not a factor

Step 19: Check divisibility by 19

19 ÷ 19 = 1 remainder 0 → 19 is a factor

Factors of 19: 1, 19 (exactly 2 factors)
Final answer:

Yes, 19 is a prime number because it has exactly two factors: 1 and 19.

Applied rules:

• Prime Definition: Must have exactly two factors

• Factor Testing: Check divisibility systematically

• Counting: Only 2 factors means it's prime

2 Exercise: Composite Identification
Exercise 2
Is 18 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
18
Factors
1, 2, 3, 6, 9, 18
Count
6 factors
Step 1: Check divisibility by 1

18 ÷ 1 = 18 remainder 0 → 1 is a factor

Step 2: Check divisibility by 2

18 ÷ 2 = 9 remainder 0 → 2 is a factor

Step 3: Check divisibility by 3

18 ÷ 3 = 6 remainder 0 → 3 is a factor

Step 4: Check divisibility by 4

18 ÷ 4 = 4 remainder 2 → 4 is not a factor

Step 5: Check divisibility by 5

18 ÷ 5 = 3 remainder 3 → 5 is not a factor

Step 6: Check divisibility by 6

18 ÷ 6 = 3 remainder 0 → 6 is a factor

Step 7: Check divisibility by 7

18 ÷ 7 = 2 remainder 4 → 7 is not a factor

Step 8: Check divisibility by 8

18 ÷ 8 = 2 remainder 2 → 8 is not a factor

Step 9: Check divisibility by 9

18 ÷ 9 = 2 remainder 0 → 9 is a factor

Step 10: Check divisibility by 10

18 ÷ 10 = 1 remainder 8 → 10 is not a factor

Step 11: Check divisibility by 11

18 ÷ 11 = 1 remainder 7 → 11 is not a factor

Step 12: Check divisibility by 12

18 ÷ 12 = 1 remainder 6 → 12 is not a factor

Step 13: Check divisibility by 13

18 ÷ 13 = 1 remainder 5 → 13 is not a factor

Step 14: Check divisibility by 14

18 ÷ 14 = 1 remainder 4 → 14 is not a factor

Step 15: Check divisibility by 15

18 ÷ 15 = 1 remainder 3 → 15 is not a factor

Step 16: Check divisibility by 16

18 ÷ 16 = 1 remainder 2 → 16 is not a factor

Step 17: Check divisibility by 17

18 ÷ 17 = 1 remainder 1 → 17 is not a factor

Step 18: Check divisibility by 18

18 ÷ 18 = 1 remainder 0 → 18 is a factor

Factors of 18: 1, 2, 3, 6, 9, 18 (6 factors total)
Final answer:

Yes, 18 is a composite number because it has more than two factors: 1, 2, 3, 6, 9, and 18.

Applied rules:

• Composite Definition: Must have more than two factors

• Factor Pairs: 1×18=18, 2×9=18, 3×6=18

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

3 Exercise: Classification
Exercise 3
From the list below, classify each number as prime or composite: 11, 14, 17, 20, 23
Definition:

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

Numbers
11, 14, 17, 20, 23
Factors
List each
Types
P, C, P, C, P
Step 1: Analyze 11

Factors: 1, 11 → Prime (exactly 2 factors)

Step 2: Analyze 14

Factors: 1, 2, 7, 14 → Composite (4 factors)

Step 3: Analyze 17

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

Step 4: Analyze 20

Factors: 1, 2, 4, 5, 10, 20 → Composite (6 factors)

Step 5: Analyze 23

Factors: 1, 23 → Prime (exactly 2 factors)

Step 6: Final classification

Prime: 11, 17, 23 | Composite: 14, 20

Prime: 11, 17, 23 | Composite: 14, 20
Final answer:

Prime numbers: 11, 17, 23 | Composite numbers: 14, 20

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

Exercise Concepts & Methods
Prime: 2 Factors | Composite: >2 Factors
Classification Rule
Exercise 1
Prime: 13 = factors 1, 13
Prime Exercise
Exercise 2
Composite: 15 = factors 1, 3, 5, 15
Composite Exercise
Exercise 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

Exercise 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 for exercises:

• 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

Exercise Solutions: 4 to 5
4 Exercise: Word Problem
Exercise 4
Tom has 29 candies. He wants to distribute them equally among some children with no candies left over. How many children can he share with? Is 29 prime or composite?
Definition:

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

Candy count
29
Factors
1, 29
Sharing options
1 child or 29 children
Step 1: Find factors of 29

Factors of 29: 1, 29

Step 2: Interpret sharing options

Option 1: Share with 1 child (29 ÷ 1 = 29 candies each)

Option 2: Share with 29 children (29 ÷ 29 = 1 candy each)

Step 3: Determine if 29 is prime or composite

Since 29 has exactly 2 factors (1 and 29), it's prime

Step 4: Answer the question

Tom can share with 1 child (giving 29 candies) or 29 children (giving 1 candy each)

29 is prime, so Tom can share with 1 or 29 children
Final answer:

Tom can share his 29 candies with 1 child (29 candies each) or 29 children (1 candy each). The number 29 is prime because it has exactly two factors: 1 and 29.

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 Exercise: Pattern Recognition
Exercise 5
Find all prime numbers between 30 and 40. List them and explain your reasoning.
Definition:

Range analysis: Systematically checking numbers within a specified range

Range
30-40
Check each
31, 32, 33, 34, 35, 36, 37, 38, 39
Primes
31, 37
Step 1: List numbers 31-39 (excluding 30 and 40)

31, 32, 33, 34, 35, 36, 37, 38, 39

Step 2: Check 31

Factors: 1, 31 → prime

Step 3: Check 32

32 = 2 × 16 → composite

Step 4: Check 33

33 = 3 × 11 → composite

Step 5: Check 34

34 = 2 × 17 → composite

Step 6: Check 35

35 = 5 × 7 → composite

Step 7: Check 36

36 = 4 × 9 → composite

Step 8: Check 37

Factors: 1, 37 → prime

Step 9: Check 38

38 = 2 × 19 → composite

Step 10: Check 39

39 = 3 × 13 → composite

Prime numbers between 30 and 40: 31, 37
Final answer:

The prime numbers between 30 and 40 are: 31 and 37

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

Exercise Concepts and Key Ideas
Prime: 2 Factors | Composite: >2 Factors
Exercise Classification
Exercise definitions:

Prime Number Exercises: Practice problems focusing on identifying numbers with exactly two factors

Composite Number Exercises: Problems focusing on identifying numbers with more than two factors

Classification Exercises: Activities that require distinguishing between prime and composite numbers

Exercise approach:
  1. Exercise 1 (Prime): Take number 13 → factors are 1 and 13 → exactly 2 factors → prime
  2. Exercise 2 (Composite): Take number 15 → factors are 1, 3, 5, 15 → 4 factors → composite
  3. Exercise 3 (Pattern): Notice that all even numbers > 2 are composite
  4. Exercise 4 (Application): Prime numbers limit how items can be shared equally
Exercise Tip 1: The first few primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29...
Exercise Tip 2: All primes > 2 are odd numbers (since even numbers > 2 are divisible by 2)
Exercise Tip 3: Primes become less frequent as numbers get larger
Exercise Tip 4: Every composite number can be broken down into prime factors
Exercise 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 Exercise: Computer security uses large prime numbers to keep passwords and messages safe.
Exercise 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!