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):

1. Write all numbers 1-30
2. Cross out 1 (not prime)
3. Circle 2, cross out all multiples of 2 (4, 6, 8, 10...)
4. Circle 3, cross out all multiples of 3 (6, 9, 12, 15...)
5. Circle 5, cross out all multiples of 5 (10, 15, 20, 25...)
6. 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! ✓