Prime Numbers

What’s a Prime?

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

Is 7 prime?

Try to divide it:

• $7 \div 2$ = 3.5 (not even)
• $7 \div 3$ = 2.33… (not even)
• $7 \div 4$ = 1.75 (not even)
• $7 \div 5$ = 1.4 (not even)
• $7 \div 6$ = 1.16… (not even)

Nothing works except 1 and 7. Yes, 7 is prime.

Is 12 prime?

• $12 \div 2$ = 6 (even!)

We found a divisor. No, 12 is not prime.

The First Few Primes

$2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31...$

Some patterns:

• 2 is the only even prime. Every other even number is divisible by 2.
• 1 is not prime. By definition, primes must be greater than 1.
• Primes get sparser as numbers get bigger, but they never stop.

Composite Numbers

Numbers that aren’t prime are called composite. They can be broken into smaller factors.

Prime Factorization

Every composite number can be written as a product of primes.

• $12 = 2 \times 2 \times 3$
• $30 = 2 \times 3 \times 5$
• $100 = 2 \times 2 \times 5 \times 5$

This breakdown is unique. There’s only one way to factor a number into primes.