Solved Exercises on Prime and Composite Numbers in Grade 4

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

Solution: Exercises 1 to 3
1 Identifying Prime Numbers
Exercise 1
Is 13 a prime number? Explain why or why not.
Definition:

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

Method:
  1. List all possible factors of the number
  2. Count the total number of factors
  3. If exactly 2 factors exist (1 and itself), it's prime
  4. If more than 2 factors exist, it's composite
Number
13
Factors
1, 13
Count
2 factors
Step 1: Test divisibility by 1

13 ÷ 1 = 13 remainder 0 → 1 is a factor

Step 2: Test divisibility by 2

13 ÷ 2 = 6 remainder 1 → 2 is not a factor

Step 3: Test divisibility by 3

13 ÷ 3 = 4 remainder 1 → 3 is not a factor

Step 4: Test divisibility by 4

13 ÷ 4 = 3 remainder 1 → 4 is not a factor

Step 5: Test divisibility by 5

13 ÷ 5 = 2 remainder 3 → 5 is not a factor

Step 6: Test divisibility by numbers up to √13 ≈ 3.6

We only need to check up to the square root of the number

13 has exactly 2 factors: 1 and 13
Final answer:

Yes, 13 is a prime number because it has exactly two factors: 1 and 13.

Applied rules:

• Prime Definition: Exactly two distinct positive divisors

• Efficiency: Only test divisibility up to the square root of the number

• Factor Counting: Prime numbers have exactly 2 factors

2 Identifying Composite Numbers
Exercise 2
Is 15 a composite number? Find all its factors.
Definition:

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

Number
15
Factors
1, 3, 5, 15
Count
4 factors
Step 1: Find all factors systematically

15 ÷ 1 = 15 → factors: 1, 15

Step 2: Continue testing

15 ÷ 2 = 7 remainder 1 → 2 is not a factor

Step 3: Continue testing

15 ÷ 3 = 5 → factors: 3, 5

Step 4: Continue testing

15 ÷ 4 = 3 remainder 3 → 4 is not a factor

Step 5: Continue testing

15 ÷ 5 = 3 (already found)

Step 6: List all unique factors

All factors of 15: 1, 3, 5, 15

15 has 4 factors: 1, 3, 5, 15
Final answer:

Yes, 15 is a composite number because it has more than two factors: 1, 3, 5, 15.

Applied rules:

• Composite Definition: More than two distinct positive divisors

• Factor Pairs: Factors come in pairs that multiply to the original number

• Systematic Approach: Test divisors in order to ensure completeness

3 Classifying Numbers
Exercise 3
From the list, classify each number as prime or composite: 7, 9, 11, 12, 17
Definition:

Classification: Systematically determining if numbers are prime or composite

Numbers
7, 9, 11, 12, 17
Factors
List each
Classifications
P, C, P, C, P
Step 1: Analyze 7

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

Step 2: Analyze 9

Factors: 1, 3, 9 → Composite (more than 2 factors)

Step 3: Analyze 11

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

Step 4: Analyze 12

Factors: 1, 2, 3, 4, 6, 12 → Composite (more than 2 factors)

Step 5: Analyze 17

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

Step 6: Final classification

7: Prime, 9: Composite, 11: Prime, 12: Composite, 17: Prime

Prime: 7, 11, 17 | Composite: 9, 12
Final answer:

Prime numbers: 7, 11, 17 | Composite numbers: 9, 12

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

Rules and methods, laws,...
Prime: Exactly 2 Factors
Prime Number Definition
Rule 1
Prime numbers have exactly 2 factors
Prime Definition
Rule 2
Composite numbers have more than 2 factors
Composite Definition
Rule 3
1 is neither prime nor composite
Special Case
Key definitions:

Prime Number: A natural number greater than 1 with exactly two distinct positive divisors: 1 and itself

Composite Number: A natural number greater than 1 with more than two distinct positive divisors

Factors: Numbers that divide evenly into another number

Complete methodology:
  1. Check if number > 1: Prime/composite only applies to numbers greater than 1
  2. Find all factors: Systematically determine all divisors
  3. Count factors: Tally the total number of factors
  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, stopping too early in divisibility tests.
Key concepts: Prime numbers are the building blocks of all numbers, every composite number can be factored into primes.
Rules to know by heart:

• 1 is neither prime nor composite

• 2 is the smallest and only even prime number

• All primes greater than 2 are odd

• The first few prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23...

• Every composite number has at least one prime factor

Solution: Exercises 4 to 5
4 Word Problem
Exercise 4
Tom has 19 marbles. He wants to arrange them in equal rows with no marbles left over. What are his options for arranging the marbles?
Definition:

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

Marble count
19
Factors
1, 19
Arrangements
1×19 or 19×1
Step 1: Determine if 19 is prime or composite

Factors of 19: 1, 19 → 19 is prime

Step 2: Find possible arrangements

Since 19 is prime, only factors are 1 and 19

Step 3: Interpret arrangements

Option 1: 1 row of 19 marbles (1×19)

Option 2: 19 rows of 1 marble each (19×1)

Step 4: Verify the solution

1 × 19 = 19 ✓, 19 × 1 = 19 ✓

Only 2 arrangements possible: 1×19 or 19×1
Final answer:

Tom can arrange the 19 marbles in only two ways: 1 row of 19 marbles or 19 rows of 1 marble each.

Applied rules:

• Prime Application: Prime numbers have limited factor pairs

• Real-world connection: Arranging objects requires understanding factors

• Verification: Confirm that arrangements use all marbles

5 Pattern Recognition
Exercise 5
Find all prime numbers between 20 and 30.
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

• Efficiency: Use divisibility rules to eliminate obvious composites

• Verification: Double-check each classification

Key Concepts: Prime and Composite Numbers
Prime: 2 Factors | Composite: >2 Factors
Fundamental Classification
Key definitions:

Prime Number: A natural number greater than 1 that has exactly two distinct positive divisors: 1 and itself

Composite Number: A natural number greater than 1 that has more than two distinct positive divisors

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

Complete methodology:
  1. Initial check: Verify the number is greater than 1
  2. Systematic factorization: Find all possible factors
  3. Count factors: Tally the total number of factors
  4. Final classification: 2 factors = prime, more than 2 = composite
Tip 1: Use divisibility rules to quickly eliminate obvious composites (even numbers, multiples of 5, etc.).
Tip 2: Remember that 2 is the only even prime number.
Tip 3: To check if a number is prime, you only need to test divisors up to its square root.
Tip 4: Practice recognizing small prime numbers (2, 3, 5, 7, 11, 13, 17, 19, 23, 29...).
Important facts: Prime numbers are infinite, every integer greater than 1 is either prime or composite, prime factorization is unique.
Real-world applications: Cryptography, computer security, number theory, mathematical proofs.
Essential rules and properties:

• The number 1 is neither prime nor composite

• 2 is the smallest prime and the only even prime

• All primes greater than 2 are odd

• Every composite number can be written as a product of primes

• Prime numbers become less frequent as numbers get larger

Questions & Answers

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

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

The definition of a prime number specifically states that it must have exactly two distinct positive divisors: 1 and itself. Since 1 only has one divisor (itself), it doesn't meet this requirement.

Additionally, mathematicians exclude 1 from being prime because it would cause problems with the fundamental theorem of arithmetic, which states that every number can be uniquely expressed as a product of primes. If 1 were prime, this uniqueness would be lost.

So 1 is in its own special category: it's neither prime nor composite.

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

Answer: You can use several strategies to make this process faster!

  • Check divisibility by small primes first: 2, 3, 5, 7, 11
  • Only check up to the square root: If n is the number, you only need to 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 113 is prime:

  • It's not even, so not divisible by 2
  • Sum of digits: 1+1+3=5, not divisible by 3
  • Doesn't end in 0 or 5, so not divisible by 5
  • √113 ≈ 10.6, so we only check up to 10
  • Check 7: 113 ÷ 7 = 16 remainder 1 → not divisible

Since we've checked all primes up to √113 and none divide 113, it's prime!

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

Answer: Yes, there are several helpful patterns for 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 20 primes to help you memorize: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71

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