Prime Numbers – The Building Blocks of All Numbers
A prime number is a whole number greater than 1 that has exactly two factors: 1 and itself. Primes are the atoms of arithmetic — every other whole number can be broken down into them.
Definition and Examples
| Number | Factors | Prime? |
|---|---|---|
| 2 | 1, 2 | Yes — and the only even prime |
| 7 | 1, 7 | Yes |
| 9 | 1, 3, 9 | No — has a factor of 3 |
| 11 | 1, 11 | Yes |
| 1 | 1 only | No — by definition, 1 is neither prime nor composite |
First 25 Prime Numbers
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
The Sieve of Eratosthenes
A 2,200-year-old method for finding all primes up to any number:
- List all numbers from 2 to your target.
- Circle 2. Cross out all multiples of 2 (4, 6, 8...).
- Circle the next uncrossed number (3). Cross out all its multiples.
- Circle 5, cross out its multiples. Continue until done.
- All circled numbers are prime.
How to Test if a Number Is Prime
Divide the number by all primes up to its square root. If none divide it exactly, it is prime.
Is 97 prime?
√97 ≈ 9.8. Test primes up to 9: 2, 3, 5, 7.
97 ÷ 2 = 48.5 ✗ 97 ÷ 3 = 32.3 ✗ 97 ÷ 5 = 19.4 ✗ 97 ÷ 7 = 13.9 ✗
No prime factor found. 97 is prime.
Interesting Facts
- 2 is the only even prime number.
- There are infinitely many prime numbers (proved by Euclid ~300 BC).
- Primes underpin internet security — RSA encryption uses huge prime numbers.
- Twin primes differ by 2 (e.g. 11 and 13, 17 and 19).
Key Takeaways
- A prime has exactly two factors: 1 and itself.
- 1 is not prime; 2 is the smallest and only even prime.
- Use the square root test to check any number efficiently.
- Primes are the foundation of prime factorization, GCF, and LCM.
Practice Questions
- List all prime numbers between 40 and 60.
- Is 91 prime? (Hint: try dividing by 7.)
- How many prime numbers are there between 1 and 30?
- Write down two twin prime pairs greater than 20.
- Explain why 1 is not considered a prime number.