Is 17 Prime or Composite?

Determine whether $17$ is prime or composite.

List what we need to check: We need to find if $17$ has any factors besides $1$ and $17$ = Check divisibility

Check divisibility by 2: $17 \div 2 = 8.5$ (not a whole number) = Not divisible by 2

Check divisibility by 3: $17 \div 3 = 5.67...$ (not a whole number) = Not divisible by 3

Check divisibility by 4: $17 \div 4 = 4.25$ (not a whole number) = Not divisible by 4

Do we need to check more?: $5 \times 5 = 25 > 17$, so we only need to check up to 4 = No more checks needed

Conclude: $17$ has no factors other than $1$ and $17$ = $17$ is prime

Answer: $17$ is a prime number because its only factors are $1$ and $17$.

Is 24 Prime or Composite?

Determine whether $24$ is prime or composite.

Check if 24 is divisible by 2: $24 \div 2 = 12$ (whole number!) = 24 is divisible by 2

Identify the factors found: Since $2 \times 12 = 24$, we found factors: $1, 2, 12, 24$ = Already more than 2 factors

Conclude: $24$ has factors other than just $1$ and $24$ = $24$ is composite

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

Finding All Prime Numbers Up to 20

List all prime numbers from $1$ to $20$.

Eliminate 1: $1$ is neither prime nor composite = Skip 1

Check 2: Factors of $2$: only $1$ and $2$ = $2$ is prime

Check 3: Factors of $3$: only $1$ and $3$ = $3$ is prime

Check 4: $4 = 2 \times 2$, has factor $2$ = $4$ is composite

Continue checking: $5$ (prime), $6 = 2 \times 3$ (composite), $7$ (prime)... = Check each number

Complete the list: Primes: $2, 3, 5, 7, 11, 13, 17, 19$ = 8 primes found

Answer: The prime numbers from $1$ to $20$ are: $2, 3, 5, 7, 11, 13, 17, 19$

Mistake: Thinking $1$ is a prime number

Why: A prime number must have exactly two different factors. The number $1$ has only one factor (itself).

Correct: $1$ is neither prime nor composite. The smallest prime number is $2$.

Mistake: Thinking all odd numbers are prime

Why: Many odd numbers have factors besides $1$ and themselves.

Correct: $9 = 3 \times 3$, $15 = 3 \times 5$, $21 = 3 \times 7$ are all odd but composite.

Mistake: Forgetting that $2$ is prime

Why: Some students think primes must be odd.

Correct: $2$ is the only even prime number. It has exactly two factors: $1$ and $2$.

Internet Security

When you shop online or log into your bank, your data is protected by encryption that uses very large prime numbers.

Security systems multiply two huge primes (hundreds of digits each). It's easy to multiply them but nearly impossible to figure out which primes were used.

Cicada Life Cycles

Some cicadas live underground for 13 or 17 years before emerging. Scientists believe they evolved prime-numbered life cycles to avoid predators.

A predator with a 2-year cycle would meet 13-year cicadas only every 26 years. A 3-year predator would meet them only every 39 years!

Q: Why is 1 not a prime number?

A: A prime number must have exactly two different factors. The number $1$ only has one factor (itself), so it doesn't meet the definition. Also, if $1$ were prime, every number would have infinitely many prime factorizations (e.g., $6 = 2 \times 3 = 1 \times 2 \times 3 = 1 \times 1 \times 2 \times 3$...).

Q: Is there a largest prime number?

A: No! There are infinitely many prime numbers. The ancient Greek mathematician Euclid proved this over 2000 years ago. As of 2024, the largest known prime has over 24 million digits!

Q: Why is 2 the only even prime?

A: Every even number greater than $2$ is divisible by $2$, so it has at least three factors ($1$, $2$, and itself). This makes all even numbers except $2$ composite.