Prime and Composite Numbers
Learn to identify prime numbers (divisible only by 1 and themselves) and composite numbers (having more than two factors).
Definition
- is prime because its only factors are and
- is composite because its factors are
Try it now
Worked Examples
Determine whether is prime or composite.
List what we need to check
We need to find if has any factors besides and → Check divisibility
Check divisibility by 2
(not a whole number) → Not divisible by 2
Check divisibility by 3
(not a whole number) → Not divisible by 3
Check divisibility by 4
(not a whole number) → Not divisible by 4
Do we need to check more?
, so we only need to check up to 4 → No more checks needed
Conclude
has no factors other than and → is prime
Answer: is a prime number because its only factors are and .
Common Mistakes
Thinking is a prime number
Why it's wrong: A prime number must have exactly two different factors. The number has only one factor (itself).
Correct: is neither prime nor composite. The smallest prime number is .
Thinking all odd numbers are prime
Why it's wrong: Many odd numbers have factors besides and themselves.
Correct: , , are all odd but composite.
Forgetting that is prime
Why it's wrong: Some students think primes must be odd.
Correct: is the only even prime number. It has exactly two factors: and .
Interactive Visual
Factor Tree
Enter a number to see its factor tree and prime factorization.
Interactive Sandbox
Expression Calculator
Try these:
History
No calculations yet
Practice Problems
15 problemsWhich of these numbers is prime?
Why It Matters
- Cryptography: Internet security and passwords rely on very large prime numbers
- Computer Science: Prime numbers help create efficient data structures
- Nature: Cicadas emerge every 13 or 17 years (both prime!) to avoid predators
- Mathematics: Every whole number can be written as a product of primes
Real World Applications
Internet Security
When you shop online or log into your bank, your data is protected by encryption that uses very large prime numbers.
Example:
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.
A simple encryption uses two small primes multiplied together: .
If someone only knows the product is , can you find the two prime factors?
Step 1: Write the mathematical expression
Find two primes that multiply to :
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.
Example:
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!
A cicada has a 17-year cycle. A predator has a 4-year cycle.
How often would the predator and cicada emerge in the same year?
Step 1: Write the mathematical expression
Find the LCM of and :
Key Takeaways
- 1A prime number has exactly two factors: and itself (examples: )
- 2A composite number has more than two factors (examples: )
- 3The number is neither prime nor composite
- 4 is the only even prime number
- 5To test if a number is prime, check divisibility by primes up to its square root
Frequently Asked Questions
Glossary
- Prime number
- A whole number greater than with exactly two factors: and itself
- Composite number
- A whole number greater than with more than two factors
- Factor
- A number that divides evenly into another number
- Divisible
- A number is divisible by another if the division results in a whole number with no remainder