What Are Prime Numbers?
A prime number is a whole number greater than 1 that has exactly two distinct factors: 1 and itself. For example, 7 is prime because it can only be divided evenly by 1 and 7. In contrast, 6 is composite because it can be divided by 1, 2, 3, and 6.
The first ten prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. Notice that 2 is the only even prime — every other even number is divisible by 2, making it composite.
Why Prime Numbers Matter
Primes are the building blocks of all integers. The Fundamental Theorem of Arithmetic states that every integer greater than 1 can be expressed as a unique product of prime numbers. This is called prime factorization. For example:
- 60 = 2 × 2 × 3 × 5
- 84 = 2 × 2 × 3 × 7
- 100 = 2 × 2 × 5 × 5
Prime factorization is essential for finding the Greatest Common Divisor (GCD) and Least Common Multiple (LCM), two concepts used constantly in fractions and number theory.
Divisibility Rules: A Quick Reference
Instead of performing long division, you can use divisibility rules to quickly determine whether a number divides evenly into another. Here are the most useful rules:
| Divisor | Rule | Example |
|---|---|---|
| 2 | Last digit is even (0, 2, 4, 6, 8) | 348 ✓ |
| 3 | Sum of digits is divisible by 3 | 123 → 1+2+3=6 ✓ |
| 4 | Last two digits form a number divisible by 4 | 1,312 → 12÷4=3 ✓ |
| 5 | Last digit is 0 or 5 | 745 ✓ |
| 6 | Divisible by both 2 and 3 | 114 ✓ |
| 9 | Sum of digits is divisible by 9 | 729 → 7+2+9=18 ✓ |
| 10 | Last digit is 0 | 980 ✓ |
How to Find Prime Factors: The Factor Tree Method
- Start with your number (e.g., 72).
- Find any two factors: 72 = 8 × 9.
- Break those down further: 8 = 2 × 4 = 2 × 2 × 2; 9 = 3 × 3.
- Stop when all branches are prime numbers.
- Result: 72 = 2³ × 3²
The Sieve of Eratosthenes
A classic algorithm for finding all primes up to a given number. Here's how it works:
- Write out all integers from 2 to your target number.
- Circle 2 (it's prime), then cross out all multiples of 2.
- Move to the next uncrossed number (3), circle it, and cross out its multiples.
- Repeat until you've processed numbers up to the square root of your target.
- All remaining circled or uncrossed numbers are prime.
Key Takeaways
- Primes have exactly two factors: 1 and themselves.
- Every composite number has a unique prime factorization.
- Divisibility rules save time and reduce arithmetic errors.
- Prime factorization unlocks GCD, LCM, and fraction simplification.
Mastering primes and divisibility gives you a powerful foundation for tackling more advanced number theory topics like modular arithmetic, cryptography, and beyond.