Solved Exercises on Prime and Composite Numbers in Pre-algebra

Master prime and composite numbers: definitions, identification methods, factorization techniques, and divisibility rules through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Identifying Prime Numbers
Exercise 1
Determine whether 17 is a prime number. Show your work using the divisibility method.
Definition:

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 positive divisors.

Prime Identification Method:
  1. Check divisibility by all prime numbers up to √n
  2. If n is divisible by any of these primes, it's composite
  3. If not divisible by any, it's prime
Number to Test
17
√17 ≈ 4.12
Test up to 4
Result
Prime
Step 1: Calculate √17

√17 ≈ 4.12, so we only need to test divisibility by primes up to 4

Primes to test: 2, 3

Step 2: Check divisibility by 2

17 ÷ 2 = 8 remainder 1 → Not divisible by 2

Since 17 is odd, it's not divisible by 2

Step 3: Check divisibility by 3

17 ÷ 3 = 5 remainder 2 → Not divisible by 3

Sum of digits: 1 + 7 = 8, not divisible by 3

Step 4: Conclusion

Since 17 is not divisible by any prime up to √17, it is prime

17 is a prime number
Final answer:

17 is a prime number because its only divisors are 1 and 17.

Applied rules:

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

• Divisibility Tests: Check division with remainder

• Prime Definition: Exactly two divisors (1 and itself)

2 Identifying Composite Numbers
Exercise 2
Determine whether 24 is a composite number. List all its factors.
Definition:

Factors: Numbers that divide evenly into another number.

Composite Number: Has more than two factors.

Number to Factor
24
Factors
1,2,3,4,6,8,12,24
Count
8 factors
Step 1: Systematic factor finding

Find factor pairs: 1×24, 2×12, 3×8, 4×6

Step 2: List all factors

From factor pairs: 1, 2, 3, 4, 6, 8, 12, 24

Step 3: Count factors

24 has 8 factors (more than 2), so it's composite

Step 4: Verify

Each factor divides 24 without remainder: 24÷1=24, 24÷2=12, etc.

24 is composite with 8 factors
Final answer:

24 is composite because it has 8 factors: 1, 2, 3, 4, 6, 8, 12, 24.

Applied rules:

• Systematic Approach: Find factor pairs to avoid missing any

• Composite Definition: More than two factors

• Verification: Each factor must divide evenly

3 Prime Factorization
Exercise 3
Find the prime factorization of 60. Express your answer in exponential form.
Definition:

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

Exponential Form: Using exponents to show repeated prime factors.

Number
60
Factor Tree
2² × 3 × 5
Result
60 = 2² × 3 × 5
Step 1: Start with smallest prime factor

60 ÷ 2 = 30 → 2 is a factor

Step 2: Continue factoring the quotient

30 ÷ 2 = 15 → 2 is a factor again

Step 3: Factor the next quotient

15 ÷ 3 = 5 → 3 is a factor

Step 4: Final factor is prime

5 is prime → 5 is our last factor

Step 5: Write in exponential form

60 = 2 × 2 × 3 × 5 = 2² × 3 × 5

60 = 2² × 3 × 5
Final answer:

The prime factorization of 60 is 2² × 3 × 5.

Applied rules:

• Division Method: Divide by smallest prime factors

• Prime Recognition: Stop when quotient is prime

• Exponential Form: Group repeated factors

Rules and methods, laws,...
Prime: Only divisors are 1 and itself
Prime Number Definition
Prime Numbers
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31...
Numbers with exactly 2 divisors
Composite Numbers
4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20...
Numbers with more than 2 divisors
Special Cases
1 is neither prime nor composite
Unique classification
Key definitions:

Prime Number: Natural number > 1 with exactly 2 divisors (1 and itself)

Composite Number: Natural number > 1 with more than 2 divisors

Prime Factorization: Expressing a number as a product of primes

Complete methodology:
  1. Identify the number: Determine if you're checking primality or finding factors
  2. Apply appropriate method: Square root method for primality, factor tree for factorization
  3. Follow systematic steps: Check divisibility systematically
  4. Verify results: Multiply factors to check accuracy
Tip 1: To check if n is prime, only test divisibility up to √n.
Tip 2: All even numbers > 2 are composite (divisible by 2).
Tip 3: Numbers ending in 5 (except 5 itself) are composite.
Tip 4: Always verify by multiplying your prime factors back together.
Common errors: Forgetting 1 is neither prime nor composite, miscounting factors, stopping factorization too early.
Exam preparation: Memorize primes up to 30, practice factor trees, master divisibility rules.
Formulas to know by heart:

• Prime Check: Test divisibility by primes ≤ √n

• Factor Count: Prime numbers have exactly 2 factors

• Even Numbers: All even numbers > 2 are composite

• Sum of Digits: If sum divisible by 3, number divisible by 3

• Ending in 5: All multiples of 5 except 5 itself are composite

Solution: Exercises 4 to 5
4 Factor Trees and Exponential Form
Exercise 4
Create a factor tree for 72 and express the prime factorization in exponential form.
Definition:

Factor Tree: A visual representation showing how a number breaks down into prime factors.

Exponential Form: Using exponents to show repeated multiplication of the same prime.

Start
72
Factor Tree
2³ × 3²
Result
72 = 2³ × 3²
Step 1: Begin with 72

72 ÷ 2 = 36 → 2 is a factor

Step 2: Factor 36

36 ÷ 2 = 18 → 2 is a factor again

Step 3: Factor 18

18 ÷ 2 = 9 → 2 is a factor again

Step 4: Factor 9

9 ÷ 3 = 3 → 3 is a factor

Step 5: Last factor is prime

3 is prime → 3 is our final factor

Step 6: Count and express exponentially

72 = 2 × 2 × 2 × 3 × 3 = 2³ × 3²

72 = 2³ × 3²
Final answer:

The prime factorization of 72 is 2³ × 3².

Applied rules:

• Systematic Division: Always divide by smallest available prime

• Counting Repetitions: Group identical prime factors

• Verification: 2³ × 3² = 8 × 9 = 72 ✓

5 Application Problem
Exercise 5
A rectangular garden has an area of 48 square meters. If both length and width must be prime numbers, what are the possible dimensions?
Definition:

Area of Rectangle: Length × Width = Area

Constraint: Both dimensions must be prime numbers

Area
48
Factor Pairs
?
Valid Pairs
None
Step 1: Find all factor pairs of 48

1×48, 2×24, 3×16, 4×12, 6×8

Step 2: Identify prime factors in pairs

Pairs with primes: 2×24 (2 is prime), 3×16 (3 is prime)

Step 3: Check if both numbers in pair are prime

2×24: 2 is prime but 24 is composite

3×16: 3 is prime but 16 is composite

Step 4: Conclusion

No factor pair consists of two prime numbers

Step 5: Verification

Prime factors of 48: 2⁴ × 3, so 48 = 2×2×2×2×3

No way to group into two primes

No valid dimensions exist
Final answer:

There are no possible dimensions where both length and width are prime numbers because no factor pair of 48 consists of two prime numbers.

Applied rules:

• Area Formula: Length × Width = Area

• Systematic Listing: Find all factor pairs

• Prime Verification: Check each factor for primality

Prime and Composite Number Properties & Techniques
n is prime ↔ divisors(n) = {1, n}
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 positive divisors.

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

Complete methodology:
  1. For Primality Testing: Check divisibility by primes up to √n
  2. For Factorization: Use systematic division by smallest primes
  3. For Verification: Multiply factors to ensure original number
  4. For Applications: Consider constraints and number properties
Tip 1: The largest prime to check for n is ≤ √n, saving significant time.
Tip 2: 2 is the only even prime number; all other even numbers are composite.
Tip 3: Use divisibility rules (2, 3, 5) to quickly eliminate candidates.
Tip 4: Always verify by multiplying your prime factors back together.
Common errors: Misclassifying 1 (neither prime nor composite), forgetting to check up to √n, stopping factorization prematurely.
Exam preparation: Memorize primes up to 30, practice divisibility rules, master factor trees.
Essential rules:

• Square Root Rule: Only test primes ≤ √n for primality

• Even Numbers: All even numbers > 2 are composite

• Sum of Digits: If sum divisible by 3, number divisible by 3

• Ending in 5: All multiples of 5 except 5 itself are composite

• Fundamental Theorem: Every composite number has a unique prime factorization

Questions & Answers

Question: I'm confused about why 1 is neither prime nor composite. Can you explain this?

Answer: This is a fundamental concept in number theory! The number 1 is special because it doesn't fit either category:

  • Not Prime: A prime number must have exactly two distinct positive divisors (1 and itself). Since 1 only has one divisor (itself), it doesn't meet the definition of prime.
  • Not Composite: A composite number must have more than two positive divisors. Since 1 only has one divisor, it doesn't meet the definition of composite either.

The definitions specifically require numbers greater than 1. This special classification ensures the Fundamental Theorem of Arithmetic works correctly, which states that every integer greater than 1 has a unique prime factorization. If 1 were considered prime, factorizations wouldn't be unique (since 1 could be included any number of times).

Question: Why do we only need to check divisibility up to √n when testing for primality? Isn't that skipping potential factors?

Answer: This is a brilliant efficiency shortcut based on mathematical logic:

  • The Logic: If n has a divisor greater than √n, it must also have a corresponding divisor less than √n.
  • Example: For n = 35, √35 ≈ 5.9. If 35 had a divisor > 5.9, say 7, then 35 ÷ 7 = 5, which is < 5.9.
  • Pairing: Divisors come in pairs: if d divides n, then n/d also divides n.

So if we've checked all possible divisors up to √n and found none, we know there can't be any larger divisors either (except n itself). This cuts the work significantly - for 100, instead of checking 98 possibilities (2 to 99), we only check 9 (2 to 9).

Question: How can I quickly determine if a large number is prime without doing extensive division?

Answer: Here are efficient strategies for primality testing:

  1. Quick Eliminations:
    • If even (except 2) → composite
    • If ends in 5 (except 5) → composite
    • If sum of digits divisible by 3 → composite
  2. Check Small Primes: Test divisibility by 2, 3, 5, 7, 11, 13 up to √n
  3. Pattern Recognition: Memorize primes up to 100 for quick recognition
  4. Calculator Assistance: Use calculator for √n and systematic division

For example, to check 97: It's not even, doesn't end in 5, sum of digits (9+7=16) isn't divisible by 3. √97 ≈ 9.8, so test 2, 3, 5, 7. None divide 97, so it's prime!