Prime Numbers

For Elementary Students

What Is a Prime Number?

A prime number is a special number that can only be divided evenly by 1 and itself.

Think about it like this: Prime numbers are like special building blocks that can't be broken into smaller whole-number pieces!

Examples of Prime Numbers

2 is prime because you can only divide it evenly by:

• 1 (2 ÷ 1 = 2)
• 2 (2 ÷ 2 = 1)

No other numbers work!

7 is prime because you can only divide it evenly by:

• 1 and 7

Try dividing 7 by 2, 3, 4, 5, or 6 — you always get a remainder!

What Is NOT Prime?

4 is NOT prime because it can be divided by:

• 1 (4 ÷ 1 = 4)
• 2 (4 ÷ 2 = 2) ← Extra number!
• 4 (4 ÷ 4 = 1)

Since 4 can be divided by more than just 1 and itself, it's NOT prime.

Composite Numbers

Numbers that are NOT prime (except 1) are called composite numbers.

Composite numbers can be broken down into smaller pieces!

Examples:

• 6 is composite (1, 2, 3, 6 all divide evenly)
• 9 is composite (1, 3, 9)
• 12 is composite (1, 2, 3, 4, 6, 12)

The Number 1 Is Special

1 is neither prime nor composite!

It's in its own special category because it only has one factor (itself).

First 10 Prime Numbers

Let's list them:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29

Try to memorize these — they're important!

Finding Prime Numbers

Is 11 prime?

Try dividing 11 by small numbers:

• 11 ÷ 2 = 5.5 (not even)
• 11 ÷ 3 = 3.67... (not even)
• 11 ÷ 4 = 2.75 (not even)

Nothing divides evenly except 1 and 11, so 11 is prime!

For Junior High Students

Precise Definition

A prime number is a whole number greater than 1 that has exactly two factors: 1 and itself.

Examples:

• 2 (factors: 1, 2)
• 3 (factors: 1, 3)
• 7 (factors: 1, 7)
• 13 (factors: 1, 13)

Composite Numbers

A composite number is a whole number greater than 1 that has more than two factors.

Examples:

• 4 (factors: 1, 2, 4)
• 6 (factors: 1, 2, 3, 6)
• 12 (factors: 1, 2, 3, 4, 6, 12)
• 15 (factors: 1, 3, 5, 15)

Special Cases to Remember

The number 1:

• Neither prime nor composite
• Only has one factor (itself)
• Special category

The number 2:

• The ONLY even prime number!
• Every other even number is divisible by 2, making them composite

The number 0:

• Not prime or composite
• Not used in this classification

Prime Numbers Up to 100

Memorize these:

• 1-10: 2, 3, 5, 7
• 11-20: 11, 13, 17, 19
• 21-30: 23, 29
• 31-40: 31, 37
• 41-50: 41, 43, 47
• 51-60: 53, 59
• 61-70: 61, 67
• 71-80: 71, 73, 79
• 81-90: 83, 89
• 91-100: 97

Total: 25 prime numbers between 1 and 100!

Testing If a Number Is Prime

Quick tests:

1. Even number? If yes (except 2), it's composite
2. Ends in 5? If yes (except 5), it's composite (divisible by 5)
3. Sum of digits divisible by 3? If yes, it's composite (divisible by 3)

Example: Is 39 prime?

• Sum of digits: 3 + 9 = 12 (divisible by 3)
• So 39 is divisible by 3: 39 ÷ 3 = 13
• 39 is composite

More Thorough Test

Test divisibility by primes up to the square root

Example: Is 29 prime?

√29 ≈ 5.4, so test primes up to 5: {2, 3, 5}

• 29 ÷ 2 = 14.5 (not even)
• 29 ÷ 3 = 9.67... (not even)
• 29 ÷ 5 = 5.8 (not even)

None divide evenly, so 29 is prime!

Example: Is 51 prime?

√51 ≈ 7.1, test: {2, 3, 5, 7}

• 51 ÷ 2 = 25.5 (not even)
• 51 ÷ 3 = 17 (divides evenly!)

51 = 3 × 17, so 51 is composite

Prime Factorization

Every composite number can be broken down into prime factors.

Example: Prime factorization of 12

      12
     /  \
    2    6
        / \
       2   3

12 = 2 × 2 × 3 (all primes!)

Example: Prime factorization of 30

      30
     /  \
    2   15
       /  \
      3    5

30 = 2 × 3 × 5

Example: Prime factorization of 100

       100
      /   \
     2    50
         /  \
        2   25
           /  \
          5    5

100 = 2 × 2 × 5 × 5 (or 2² × 5²)

Why Prime Factorization Matters

Used for:

• Finding GCD (Greatest Common Divisor)
• Finding LCM (Least Common Multiple)
• Simplifying fractions
• Understanding number properties

Interesting Prime Facts

Infinite primes: There are infinitely many prime numbers!

Twin primes: Prime pairs that differ by 2 (like 11 and 13, or 17 and 19)

Mersenne primes: Primes in the form 2ⁿ − 1 (like 3, 7, 31, 127)

Largest known prime: Has millions of digits! (Found using computers)

Real-Life Uses

Cryptography: Internet security uses prime numbers!

Computer science: Prime numbers help with hashing and data structures

Mathematics: Foundation for number theory

Practice