Solved Exercises on Prime and Composite Numbers

Master prime and composite numbers: identifying primes, finding composites, and understanding factors through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Identifying Prime Numbers
Exercise 1
Determine if 13 is a prime number and explain why.
Definition:

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

Prime identification method:
  1. List all possible factors of the number
  2. Count the total number of factors
  3. If there are exactly 2 factors, it's prime
  4. If there are more than 2 factors, it's composite
Number
13
Factors
1, 13
Type
Prime
Step 1: List all factors of 13

We need to find all numbers that divide 13 evenly (with no remainder)

13 ÷ 1 = 13 (remainder 0) ✓

13 ÷ 2 = 6 remainder 1 (not a factor)

13 ÷ 3 = 4 remainder 1 (not a factor)

13 ÷ 4 = 3 remainder 1 (not a factor)

13 ÷ 5 = 2 remainder 3 (not a factor)

13 ÷ 6 = 2 remainder 1 (not a factor)

13 ÷ 7 = 1 remainder 6 (not a factor)

13 ÷ 8 = 1 remainder 5 (not a factor)

13 ÷ 9 = 1 remainder 4 (not a factor)

13 ÷ 10 = 1 remainder 3 (not a factor)

13 ÷ 11 = 1 remainder 2 (not a factor)

13 ÷ 12 = 1 remainder 1 (not a factor)

Step 2: Count the factors

Only 1 and 13 divide 13 evenly

Total factors: 2

Step 3: Determine if prime or composite

Since 13 has exactly 2 factors (1 and 13), it is a prime number

13 is PRIME
Final answer:

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

Applied rules:

• Prime Definition: Exactly two factors: 1 and the number itself

• Factor Testing: Check divisibility by all numbers less than the square root

• Counting Factors: Prime numbers have exactly 2 factors

2 Identifying Composite Numbers
Exercise 2
Determine if 18 is a composite number and list all its factors.
Definition:

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

Number
18
Factors
1,2,3,6,9,18
Type
Composite
Step 1: Find all factors of 18

We need to find all numbers that divide 18 evenly (with no remainder)

18 ÷ 1 = 18 (remainder 0) ✓

18 ÷ 2 = 9 (remainder 0) ✓

18 ÷ 3 = 6 (remainder 0) ✓

18 ÷ 6 = 3 (remainder 0) ✓

18 ÷ 9 = 2 (remainder 0) ✓

18 ÷ 18 = 1 (remainder 0) ✓

Step 2: List all factors in order

Factors of 18: 1, 2, 3, 6, 9, 18

Step 3: Count the factors

Total factors: 6 (more than 2)

Step 4: Determine if prime or composite

Since 18 has more than 2 factors, it is a composite number

18 is COMPOSITE with factors: 1, 2, 3, 6, 9, 18
Final answer:

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

Applied rules:

• Composite Definition: More than two factors

• Factor Pairs: When you find one factor, you also find its pair

• Systematic Approach: Check divisors up to the square root of the number

3 Finding Prime or Composite
Exercise 3
Determine if 23 is prime or composite. Show your work.
Definition:

Prime number: Has exactly two factors (1 and itself)

Number
23
Factors
1, 23
Type
Prime
Step 1: Test divisibility by small prime numbers

We only need to check divisors up to √23 ≈ 4.8, so check 2, 3, and 4

23 ÷ 2 = 11 remainder 1 (not divisible)

23 ÷ 3 = 7 remainder 2 (not divisible)

23 ÷ 4 = 5 remainder 3 (not divisible)

Step 2: Check if any other numbers divide 23

Since we've checked all potential divisors up to √23, we can stop

Step 3: List all factors

Only 1 and 23 divide 23 evenly

Factors: 1, 23

Step 4: Determine if prime or composite

Since 23 has exactly 2 factors, it is prime

23 is PRIME
Final answer:

23 is a prime number because it has exactly two factors: 1 and 23

Applied rules:

• Square Root Rule: Only check divisors up to √n

• Efficient Testing: Check small primes first (2, 3, 5, 7...)

• Prime Definition: Exactly two factors confirms primality

Prime and Composite Numbers Summary & Methods
\(\text{Prime}: \text{Exactly 2 factors (1 and itself)}\)
Prime Number Definition
Prime
\(2, 3, 5, 7, 11, 13...\)
Exactly 2 factors
Composite
\(4, 6, 8, 9, 10, 12...\)
More than 2 factors
Neither
\(1\)
Only 1 factor
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

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

Divisor: A number that can divide another number without remainder

Prime Factorization: Expressing a number as a product of prime numbers

Factor Pair: Two numbers that multiply to give the original number

Prime/composite identification methods:
  1. List all factors: Find all numbers that divide the target number evenly
  2. Count factors: If exactly 2, it's prime; if more than 2, it's composite
  3. Use divisibility rules: Check for quick division by small numbers
  4. Square root shortcut: Only check divisors up to √n
  5. Check small primes: Test divisibility by 2, 3, 5, 7, etc.
  6. Verify result: Confirm your classification makes sense
Tip 1: All even numbers greater than 2 are composite (divisible by 2).
Tip 2: Numbers ending in 5 (except 5 itself) are composite (divisible by 5).
Tip 3: If a number is divisible by 3, add digits and check if sum is divisible by 3.
Tip 4: Use the square root rule to save time when testing large numbers.
Common errors: Forgetting that 1 is neither prime nor composite, thinking all odd numbers are prime, not checking enough divisors, miscounting factors, confusing factors with multiples.
Key notes: 2 is the only even prime number, prime numbers are building blocks for all other numbers, memorize the first 10-15 prime numbers.
Essential facts:

• Smallest prime: 2

• Smallest composite: 4

• 1 is neither prime nor composite

• 2 is the only even prime number

• All primes > 3 are of the form 6n±1

Solution: Exercises 4 to 5
4 Prime Factorization
Exercise 4
Find the prime factorization of 24 and identify if it's prime or composite.
Definition:

Prime factorization: Expressing a number as a product of prime numbers

Number
24
Prime Factors
2³ × 3
Type
Composite
Step 1: Start with the smallest prime factor

24 is even, so divide by 2: 24 ÷ 2 = 12

12 is even, so divide by 2: 12 ÷ 2 = 6

6 is even, so divide by 2: 6 ÷ 2 = 3

3 is prime, so divide by 3: 3 ÷ 3 = 1

Step 2: Write the prime factorization

24 = 2 × 2 × 2 × 3 = 2³ × 3

Step 3: Find all factors of 24

Using the prime factorization: 2³ × 3¹

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

Step 4: Determine if prime or composite

Since 24 has more than 2 factors (8 total), it is composite

24 = 2³ × 3 (Composite)
Final answer:

The prime factorization of 24 is 2³ × 3, and 24 is composite because it has more than two factors

Applied rules:

• Prime Factorization: Repeatedly divide by smallest prime factors

• Factor Count: For prime factorization a^m × b^n, factors = (m+1)(n+1)

• Composite Confirmation: More than 2 factors means composite

5 Comparison Problem
Exercise 5
Which number is prime: 37 or 45? Explain your reasoning.
Definition:

Comparison: Testing each number separately to determine primality

37 Test
Prime
45 Test
Composite
Answer
37 is prime
Step 1: Test 37 for primality

Check divisibility by primes up to √37 ≈ 6.1 (so check 2, 3, 5)

37 ÷ 2 = 18 remainder 1 (not divisible)

37 ÷ 3 = 12 remainder 1 (not divisible)

37 ÷ 5 = 7 remainder 2 (not divisible)

37 has exactly 2 factors: 1 and 37 → PRIME

Step 2: Test 45 for primality

Check divisibility by primes up to √45 ≈ 6.7 (so check 2, 3, 5)

45 ÷ 2 = 22 remainder 1 (not divisible by 2)

45 ÷ 3 = 15 remainder 0 (divisible by 3!) → COMPOSITE

45 = 3 × 15, so factors include: 1, 3, 5, 9, 15, 45

Step 3: Compare results

37: Prime (2 factors)

45: Composite (6 factors)

37 is prime, 45 is composite
Final answer:

37 is prime because it has exactly two factors (1 and 37), while 45 is composite because it has more than two factors (1, 3, 5, 9, 15, 45)

Applied rules:

• Systematic Testing: Check divisibility by primes up to √n

• Quick Divisibility: For 3, add digits (4+5=9, divisible by 3)

• Comparison Logic: Test each number individually before comparing

Complete Guide to Prime and Composite Numbers
\(\text{Prime}: \text{Exactly 2 factors}\)
Prime Definition
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

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

Prime Factorization: Expressing a number as a product of prime numbers

Factor Tree: A visual representation of prime factorization

Divisibility Rules: Quick methods to check if one number divides another

Factor Pair: Two numbers that multiply to give the original number

Prime/composite identification methodology:
  1. Initial assessment: Check if number is 1 (neither prime nor composite)
  2. Quick elimination: Even numbers > 2 and multiples of 5 are composite
  3. List all factors: Find all numbers that divide the target evenly
  4. Count factors: Exactly 2 = prime, more than 2 = composite
  5. Use shortcuts: Apply divisibility rules and square root method
  6. Verify: Double-check your classification
Tip 1: The first 10 prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.
Tip 2: To find factors efficiently, test divisors up to the square root of the number.
Tip 3: If a number ends in 0, 2, 4, 6, or 8, it's divisible by 2 (composite if > 2).
Tip 4: For divisibility by 3, add all digits and check if sum is divisible by 3.
Tip 5: Numbers ending in 5 (except 5 itself) are divisible by 5 and thus composite.
Tip 6: Always verify your answer by checking that all factors divide the number evenly.
Common errors: Forgetting that 1 is neither prime nor composite, thinking all odd numbers are prime, not checking enough divisors, miscounting factors, confusing factors with multiples, assuming larger numbers are always composite.
Exam preparation: Memorize the first 20 prime numbers, practice divisibility rules, master the square root shortcut, understand that 2 is the only even prime, practice prime factorization, work on comparison problems.
Essential facts:

• Smallest prime: 2 (also the only even prime)

• Smallest composite: 4

• 1 is neither prime nor composite

• All primes greater than 3 are of the form 6n±1

• Prime factorization is unique for each number

• Every composite number has at least one prime factor ≤ √n

Prime and Composite Visualization
Exercise 6: Prime vs Composite Distribution
Compare the distribution of prime and composite numbers:
Numbers 1-20: Count primes and composites
Numbers 21-40: Count primes and composites
Numbers 41-60: Count primes and composites

Analysis: The chart shows how prime numbers become less frequent as numbers get larger.

  • Primes 1-20: 2, 3, 5, 7, 11, 13, 17, 19 (8 primes)
  • Primes 21-40: 23, 29, 31, 37 (4 primes)
  • Primes 41-60: 41, 43, 47, 53, 59 (5 primes)

Questions & Answers

Question: Why is 1 neither prime nor composite?

Answer: The number 1 is special because it has only one factor: itself (1). Let's look at the definitions:

  • Prime number: Must have exactly 2 factors (1 and itself)
  • Composite number: Must have more than 2 factors

Since 1 only has 1 factor, it doesn't meet the criteria for either category. This is why 1 gets its own special classification as "neither prime nor composite."

This definition helps keep mathematical theorems and properties consistent, particularly in prime factorization where every number has a unique prime factorization.

Question: Is 2 a prime number? I thought all even numbers were composite.

Answer: Yes, 2 is definitely a prime number! You're right that 2 is even, but it's a special case:

  • 2 is divisible by 1: 2 ÷ 1 = 2
  • 2 is divisible by 2: 2 ÷ 2 = 1
  • 2 has exactly 2 factors: 1 and 2

So 2 meets the definition of a prime number perfectly. It's actually the smallest prime number and the only even prime number.

The general rule is: "All even numbers greater than 2 are composite," which means 2 is the exception to that rule.

This is important because 2 is needed for prime factorization of even numbers!

Question: How can I quickly tell if a large number is prime without listing all factors?

Answer: Here are efficient methods to test primality for larger numbers:

  1. Check divisibility rules: If the number is even, ends in 5, or digit sum is divisible by 3, it's composite
  2. Square root rule: Only test divisors up to √n (where n is your number)
  3. Test small primes: Check divisibility by 2, 3, 5, 7, 11, 13, etc.
  4. Stop early: If you find any divisor other than 1 and the number itself, it's composite

Example for 47: √47 ≈ 6.9, so test 2, 3, 5:

  • 47 is not even → not divisible by 2
  • 4+7=11, not divisible by 3 → not divisible by 3
  • Doesn't end in 5 → not divisible by 5

Since no divisors were found, 47 is prime!

Question: What's the difference between factors and multiples?

Answer: Here's the key difference:

Factors: Numbers that divide evenly into another number

  • Example: Factors of 12 are 1, 2, 3, 4, 6, 12
  • Factors are always smaller than or equal to the number
  • There's a finite number of factors for any number

Multiples: Results of multiplying a number by whole numbers

  • Example: Multiples of 3 are 3, 6, 9, 12, 15, 18...
  • They get larger as you continue the sequence
  • There are infinitely many multiples for any number

Think of it this way: factors go INTO a number, multiples come FROM a number!

When determining if a number is prime or composite, we focus on its factors.

Question: What's the best way to practice identifying prime and composite numbers?

Answer: Here's an effective practice strategy:

  1. Memorize small primes: Know the first 15-20 prime numbers by heart
  2. Practice divisibility rules: Master quick checks for 2, 3, 5, 9, 10
  3. Use the square root method: Practice testing divisors only up to √n
  4. Work systematically: Always check small primes first (2, 3, 5, 7...)
  5. Create lists: Make charts of primes and composites in different ranges
  6. Take timed challenges: Practice identifying numbers quickly

Daily practice with 10-15 numbers will significantly improve your speed and accuracy!

Remember: Understanding the concepts deeply is more important than just memorizing numbers!