Solved Exercises on Prime and Composite Numbers Anchor Chart

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

Prime and Composite Numbers Anchor Chart
Prime and Composite Numbers Anchor Chart
Prime Numbers
2
3
5
7
11
13
17
19
23
29
31
37
41
43
47
Numbers with exactly 2 factors: 1 and itself
Composite Numbers
4
6
8
9
10
12
14
15
16
18
20
21
22
24
25
Numbers with more than 2 factors
Special Case
1
Neither prime nor composite
Rule: Exactly 2 factors = Prime
Rule: More than 2 factors = Composite
Tip: 2 is the only even prime
Solution: Exercises 1 to 3
1 Identifying Prime Numbers
Exercise 1
Using the anchor chart, identify if 17 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. Check if the number is in the prime numbers section of the anchor chart
  2. List all possible factors of the number
  3. Count the total number of factors
  4. If there are exactly 2 factors, it's prime
Number
17
Factors
1, 17
Type
Prime
Step 1: Locate 17 on the anchor chart

Looking at the anchor chart, 17 is listed in the "Prime Numbers" section

Step 2: List all factors of 17

17 ÷ 1 = 17 (remainder 0) ✓

17 ÷ 2 = 8 remainder 1 (not a factor)

17 ÷ 3 = 5 remainder 2 (not a factor)

17 ÷ 4 = 4 remainder 1 (not a factor)

17 ÷ 5 = 3 remainder 2 (not a factor)

17 ÷ 6 = 2 remainder 5 (not a factor)

17 ÷ 7 = 2 remainder 3 (not a factor)

17 ÷ 8 = 2 remainder 1 (not a factor)

17 ÷ 9 = 1 remainder 8 (not a factor)

17 ÷ 10 = 1 remainder 7 (not a factor)

17 ÷ 11 = 1 remainder 6 (not a factor)

17 ÷ 12 = 1 remainder 5 (not a factor)

17 ÷ 13 = 1 remainder 4 (not a factor)

17 ÷ 14 = 1 remainder 3 (not a factor)

17 ÷ 15 = 1 remainder 2 (not a factor)

17 ÷ 16 = 1 remainder 1 (not a factor)

Step 3: Count the factors

Only 1 and 17 divide 17 evenly

Total factors: 2

Step 4: Determine if prime or composite

Since 17 has exactly 2 factors (1 and 17), it is a prime number

17 is PRIME
Final answer:

Yes, 17 is a prime number because it has exactly two factors: 1 and 17

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
Using the anchor chart, determine if 24 is a composite number and list all its factors.
Definition:

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

Number
24
Factors
1,2,3,4,6,8,12,24
Type
Composite
Step 1: Locate 24 on the anchor chart

Looking at the anchor chart, 24 is listed in the "Composite Numbers" section

Step 2: Find all factors of 24

We need to find all numbers that divide 24 evenly (with no remainder)

24 ÷ 1 = 24 (remainder 0) ✓

24 ÷ 2 = 12 (remainder 0) ✓

24 ÷ 3 = 8 (remainder 0) ✓

24 ÷ 4 = 6 (remainder 0) ✓

24 ÷ 6 = 4 (remainder 0) ✓

24 ÷ 8 = 3 (remainder 0) ✓

24 ÷ 12 = 2 (remainder 0) ✓

24 ÷ 24 = 1 (remainder 0) ✓

Step 3: List all factors in order

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

Step 4: Count the factors

Total factors: 8 (more than 2)

Step 5: Determine if prime or composite

Since 24 has more than 2 factors, it is a composite number

24 is COMPOSITE with factors: 1, 2, 3, 4, 6, 8, 12, 24
Final answer:

Yes, 24 is a composite number because it has more than two factors: 1, 2, 3, 4, 6, 8, 12, 24

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
Using the anchor chart, determine if 29 is prime or composite. Show your work.
Definition:

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

Number
29
Factors
1, 29
Type
Prime
Step 1: Locate 29 on the anchor chart

Looking at the anchor chart, 29 is listed in the "Prime Numbers" section

Step 2: Test divisibility by small prime numbers

We only need to check divisors up to √29 ≈ 5.4, so check 2, 3, and 5

29 ÷ 2 = 14 remainder 1 (not divisible)

29 ÷ 3 = 9 remainder 2 (not divisible)

29 ÷ 5 = 5 remainder 4 (not divisible)

Step 3: Check if any other numbers divide 29

Since we've checked all potential divisors up to √29, we can stop

Step 4: List all factors

Only 1 and 29 divide 29 evenly

Factors: 1, 29

Step 5: Determine if prime or composite

Since 29 has exactly 2 factors, it is prime

29 is PRIME
Final answer:

29 is a prime number because it has exactly two factors: 1 and 29

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. Use anchor chart: Check if number is listed in the appropriate section
  2. List all factors: Find all numbers that divide the target number evenly
  3. Count factors: If exactly 2, it's prime; if more than 2, it's composite
  4. Use divisibility rules: Check for quick division by small numbers
  5. Square root shortcut: Only check divisors up to √n
  6. Check small primes: Test divisibility by 2, 3, 5, 7, etc.
  7. 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
Using the anchor chart, find the prime factorization of 36 and identify if it's prime or composite.
Definition:

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

Number
36
Prime Factors
2² × 3²
Type
Composite
Step 1: Locate 36 on the anchor chart

Looking at the anchor chart, 36 is not listed in the "Prime Numbers" section, so it's composite

Step 2: Start with the smallest prime factor

36 is even, so divide by 2: 36 ÷ 2 = 18

18 is even, so divide by 2: 18 ÷ 2 = 9

9 is not even, so try next prime (3): 9 ÷ 3 = 3

3 is prime, so divide by 3: 3 ÷ 3 = 1

Step 3: Write the prime factorization

36 = 2 × 2 × 3 × 3 = 2² × 3²

Step 4: Find all factors of 36

Using the prime factorization: 2² × 3²

Factors: 1, 2, 3, 4, 6, 9, 12, 18, 36

Step 5: Determine if prime or composite

Since 36 has more than 2 factors (9 total), it is composite

36 = 2² × 3² (Composite)
Final answer:

The prime factorization of 36 is 2² × 3², and 36 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
Using the anchor chart, which number is prime: 47 or 51? Explain your reasoning.
Definition:

Comparison: Testing each number separately to determine primality

47 Test
Prime
51 Test
Composite
Answer
47 is prime
Step 1: Locate 47 and 51 on the anchor chart

Looking at the anchor chart, 47 is listed in the "Prime Numbers" section

51 is not listed in the "Prime Numbers" section, so we need to check it

Step 2: Test 47 for primality

Check divisibility by primes up to √47 ≈ 6.9 (so check 2, 3, 5)

47 ÷ 2 = 23 remainder 1 (not divisible)

47 ÷ 3 = 15 remainder 2 (not divisible)

47 ÷ 5 = 9 remainder 2 (not divisible)

47 has exactly 2 factors: 1 and 47 → PRIME

Step 3: Test 51 for primality

Check divisibility by primes up to √51 ≈ 7.1 (so check 2, 3, 5, 7)

51 ÷ 2 = 25 remainder 1 (not divisible by 2)

51 ÷ 3 = 17 remainder 0 (divisible by 3!) → COMPOSITE

51 = 3 × 17, so factors include: 1, 3, 17, 51

Step 4: Compare results

47: Prime (2 factors)

51: Composite (4 factors)

47 is prime, 51 is composite
Final answer:

47 is prime because it has exactly two factors (1 and 47), while 51 is composite because it has more than two factors (1, 3, 17, 51)

Applied rules:

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

• Quick Divisibility: For 3, add digits (5+1=6, 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. Reference anchor chart: Check if number is listed in the chart
  2. Initial assessment: Check if number is 1 (neither prime nor composite)
  3. Quick elimination: Even numbers > 2 and multiples of 5 are composite
  4. List all factors: Find all numbers that divide the target evenly
  5. Count factors: Exactly 2 = prime, more than 2 = composite
  6. Use shortcuts: Apply divisibility rules and square root method
  7. 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!