Prime and Composite Numbers
Identify prime and composite numbers
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Prime and Composite Numbers
What are Prime Numbers?
A prime number is a number that has EXACTLY TWO factors: 1 and itself.
Examples of prime numbers:
- 2 (factors: 1, 2) ✓
- 3 (factors: 1, 3) ✓
- 5 (factors: 1, 5) ✓
- 7 (factors: 1, 7) ✓
- 11 (factors: 1, 11) ✓
Key fact: Prime numbers can ONLY be divided evenly by 1 and themselves!
What are Composite Numbers?
A composite number is a number that has MORE than two factors.
Examples of composite numbers:
- 4 (factors: 1, 2, 4) ✓ - three factors
- 6 (factors: 1, 2, 3, 6) ✓ - four factors
- 8 (factors: 1, 2, 4, 8) ✓ - four factors
- 9 (factors: 1, 3, 9) ✓ - three factors
- 10 (factors: 1, 2, 5, 10) ✓ - four factors
Key fact: Composite numbers can be divided evenly by numbers OTHER than just 1 and themselves!
The Special Numbers
The Number 1
1 is NEITHER prime nor composite!
Why? Because 1 only has ONE factor (itself).
- Prime needs exactly TWO factors
- Composite needs MORE than two factors
- 1 has only ONE factor
1 is special and gets its own category!
The Number 2
2 is the ONLY even prime number!
- Factors of 2: 1, 2 (exactly two) ✓ Prime!
- All other even numbers are composite (divisible by 2)
2 is the smallest prime number!
Prime Numbers 1-100
Here are all the prime numbers from 1 to 100:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97
That's 25 prime numbers from 1 to 100!
Notice:
- 2 is the only even prime
- All other primes are odd
- They get more spread out as numbers get bigger
How to Tell if a Number is Prime
Method: Test for factors
Example: Is 17 prime?
Step 1: Try dividing by 2 17 ÷ 2 = 8.5 ✗ (not evenly)
Step 2: Try dividing by 3 17 ÷ 3 = 5.67... ✗ (not evenly)
Step 3: Try dividing by 4 17 ÷ 4 = 4.25 ✗ (not evenly)
Step 4: Try dividing by 5 17 ÷ 5 = 3.4 ✗ (not evenly)
We can stop here! (Why? If there are no factors by now, there won't be any)
17 is PRIME! ✓ (only factors are 1 and 17)
Shortcut: When to Stop Testing
You only need to test up to the square root of the number!
Example: Is 29 prime?
Square root of 29 ≈ 5.4
Test divisors: 2, 3, 4, 5 (stop at 5!)
- 29 ÷ 2 = 14.5 ✗
- 29 ÷ 3 = 9.67... ✗
- 29 ÷ 4 = 7.25 ✗
- 29 ÷ 5 = 5.8 ✗
29 is PRIME! ✓
Testing Small Numbers
Is 2 prime? Factors: 1, 2 Yes! ✓
Is 3 prime? Factors: 1, 3 Yes! ✓
Is 4 prime? Factors: 1, 2, 4 No - it's composite! (has more than 2 factors)
Is 5 prime? Factors: 1, 5 Yes! ✓
Is 6 prime? Factors: 1, 2, 3, 6 No - it's composite!
Is 7 prime? Factors: 1, 7 Yes! ✓
Is 8 prime? Factors: 1, 2, 4, 8 No - it's composite!
Is 9 prime? Factors: 1, 3, 9 No - it's composite!
Is 10 prime? Factors: 1, 2, 5, 10 No - it's composite!
Quick Tests for Composite Numbers
Divisible by 2? (even number) → Composite!
- 24, 68, 102 are all composite
Divisible by 3? (digit sum divisible by 3) → Composite!
- 27 → 2+7=9, 9÷3=3 → Composite!
Divisible by 5? (ends in 0 or 5) → Composite!
- 25, 35, 45, 50 are all composite
Exception: 2, 3, and 5 themselves ARE prime!
Prime or Composite Chart
| Number | Factors | Prime or Composite? |
|---|---|---|
| 1 | 1 | Neither |
| 2 | 1, 2 | Prime ✓ |
| 3 | 1, 3 | Prime ✓ |
| 4 | 1, 2, 4 | Composite |
| 5 | 1, 5 | Prime ✓ |
| 6 | 1, 2, 3, 6 | Composite |
| 7 | 1, 7 | Prime ✓ |
| 8 | 1, 2, 4, 8 | Composite |
| 9 | 1, 3, 9 | Composite |
| 10 | 1, 2, 5, 10 | Composite |
| 11 | 1, 11 | Prime ✓ |
| 12 | 1, 2, 3, 4, 6, 12 | Composite |
Why Do We Care?
Prime numbers are the "building blocks" of all numbers!
Every composite number can be broken down into prime factors.
Example: 12 = 2 × 2 × 3
- 12 is composite
- It's made up of the primes 2 and 3
This is called prime factorization (you'll learn more later)!
Common Mistakes
❌ Mistake 1: Thinking 1 is prime 1 is NEITHER prime nor composite! ✓
❌ Mistake 2: Thinking 2 is composite because it's even 2 IS prime (the only even prime)! ✓
❌ Mistake 3: Thinking all odd numbers are prime 9, 15, 21, 25, 27 are odd but composite! ✓
Memory Tricks
Remember the first few primes: "2, 3, 5, 7, 11, 13 went to 17 to buy 19, 23 happy 29 year-olds 31 books..."
Or just remember: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29
Identifying Strategy
To determine if a number is prime or composite:
Step 1: Is it 1? → Neither Step 2: Is it 2? → Prime Step 3: Is it even? → Composite Step 4: Try dividing by small primes (2, 3, 5, 7...) Step 5: If no factors found → Prime!
Real-World Applications
Cryptography (Internet Security): Prime numbers are used to keep your passwords and credit card info safe online!
Nature: Some cicadas come out every 13 or 17 years (both prime numbers) - this helps them survive!
Twin Primes
Twin primes are prime numbers that differ by 2.
Examples:
- 3 and 5 (5 - 3 = 2) ✓
- 5 and 7 (7 - 5 = 2) ✓
- 11 and 13 (13 - 11 = 2) ✓
- 17 and 19 ✓
- 29 and 31 ✓
Cool pattern! ✓
The Sieve of Eratosthenes
An ancient method to find all primes up to a certain number!
How it works (for numbers 1-30):
- Write all numbers 1-30
- Cross out 1 (not prime)
- Circle 2, cross out all multiples of 2 (4, 6, 8, 10...)
- Circle 3, cross out all multiples of 3 (6, 9, 12, 15...)
- Circle 5, cross out all multiples of 5 (10, 15, 20, 25...)
- Continue...
Numbers left circled are all prime! ✓
Practice Examples
Example 1: Is 13 prime or composite?
- Test: 13 ÷ 2 = 6.5 ✗
- Test: 13 ÷ 3 = 4.33... ✗
- Factors: only 1 and 13
- Answer: Prime! ✓
Example 2: Is 15 prime or composite?
- Factors: 1, 3, 5, 15
- Has more than 2 factors
- Answer: Composite! ✓
Example 3: Is 23 prime or composite?
- Test: 23 ÷ 2 = 11.5 ✗
- Test: 23 ÷ 3 = 7.67... ✗
- Test: 23 ÷ 5 = 4.6 ✗
- Factors: only 1 and 23
- Answer: Prime! ✓
Quick Reference
Prime Number:
- Exactly 2 factors (1 and itself)
- Can't be divided evenly by any other numbers
- Examples: 2, 3, 5, 7, 11, 13...
Composite Number:
- More than 2 factors
- Can be divided evenly by numbers other than 1 and itself
- Examples: 4, 6, 8, 9, 10, 12...
Neither:
- The number 1 (only has 1 factor)
Summary
Every whole number greater than 1 is either prime or composite!
Prime: Building blocks (can't be broken down further) Composite: Made up of primes (can be broken down)
Remember:
- 1 is neither
- 2 is the only even prime
- All even numbers (except 2) are composite
- Not all odd numbers are prime!
To test: Find all factors. If there are exactly 2 (1 and itself), it's prime! ✓
📚 Practice Problems
1Problem 1easy
❓ Question:
Is 7 a prime number or a composite number?
💡 Show Solution
To determine if 7 is prime, check if it has exactly two factors: 1 and itself.
Factors of 7: 1 × 7 = 7
That's it! 7 only has two factors: 1 and 7.
Answer: 7 is a PRIME number.
2Problem 2easy
❓ Question:
Is 12 a prime number or a composite number?
💡 Show Solution
Check the factors of 12.
Factors of 12: 1 × 12 = 12 2 × 6 = 12 3 × 4 = 12
12 has more than two factors: 1, 2, 3, 4, 6, 12
Answer: 12 is a COMPOSITE number.
3Problem 3medium
❓ Question:
List all prime numbers between 20 and 30.
💡 Show Solution
Check each number between 20 and 30:
20: Composite (divisible by 2, 4, 5, 10) 21: Composite (divisible by 3, 7) 22: Composite (divisible by 2, 11) 23: Prime (only divisible by 1 and 23) 24: Composite (divisible by 2, 3, 4, 6, 8, 12) 25: Composite (divisible by 5) 26: Composite (divisible by 2, 13) 27: Composite (divisible by 3, 9) 28: Composite (divisible by 2, 4, 7, 14) 29: Prime (only divisible by 1 and 29) 30: Composite (divisible by 2, 3, 5, 6, 10, 15)
Answer: 23 and 29
4Problem 4medium
❓ Question:
Is 1 a prime number, composite number, or neither?
💡 Show Solution
Let's check the definition:
Prime number: Has EXACTLY two factors (1 and itself) Composite number: Has MORE than two factors
Factors of 1: Just 1 (because 1 × 1 = 1)
1 only has ONE factor, not two.
Answer: 1 is NEITHER prime nor composite. It's a special number!
5Problem 5hard
❓ Question:
Explain why 51 is NOT a prime number.
💡 Show Solution
To prove 51 is not prime, we need to find a factor other than 1 and 51.
Check if 51 is divisible by small primes:
- Divisible by 2? No (51 is odd)
- Divisible by 3? Let's check: 5 + 1 = 6, which is divisible by 3, so YES!
51 ÷ 3 = 17
So 51 = 3 × 17
Factors of 51: 1, 3, 17, 51
Answer: 51 is composite because it has factors other than 1 and itself. Specifically, 51 = 3 × 17.
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