What is Prime Factorization?
Prime factorization is the process of expressing a composite number as a product of prime numbers. Every positive integer greater than 1 can be uniquely represented as a product of prime factors — this is known as the Fundamental Theorem of Arithmetic.
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. The first few prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29...
Quick Example: Prime Factorization of 60
60 = 2 × 2 × 3 × 5 = 2² × 3 × 5
The prime factors of 60 are 2, 3, and 5. We can verify: 2 × 2 × 3 × 5 = 4 × 15 = 60 ✓
The Division Method (Step-by-Step)
The division method is the most practical way to find prime factors by hand. Here's how humans do it:
How the Division Method Works
Start with the smallest prime (2)
Check if your number is even. If yes, divide by 2.
Keep dividing by the same prime
Continue dividing by 2 until it no longer divides evenly.
Move to the next prime (3, then 5, 7, 11...)
Try dividing by the next prime number and repeat.
Stop when you reach 1
The prime factorization is the product of all divisors used.
Worked Example: Find prime factors of 84
| 2 | 84 ÷ 2 = 42 (84 is even) |
| 2 | 42 ÷ 2 = 21 (42 is even) |
| 3 | 21 ÷ 3 = 7 (21 is divisible by 3) |
| 7 | 7 ÷ 7 = 1 (7 is prime) |
Result: 84 = 2 × 2 × 3 × 7 = 2² × 3 × 7
Prime vs. Composite Numbers
Prime Numbers
Divisible only by 1 and themselves. Cannot be factored further.
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37...
Composite Numbers
Have more than two factors. Can be expressed as products of primes.
4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20...
Applications of Prime Factorization
Finding GCD and LCM
Prime factorization makes it easy to find the Greatest Common Divisor and Least Common Multiple of two or more numbers.
Simplifying Fractions
Factor both numerator and denominator to find common factors that can be cancelled out.
Cryptography (RSA)
Modern encryption relies on the difficulty of factoring very large numbers into primes.
Simplifying Square Roots
Use prime factors to simplify radicals:
Quick Reference: First 20 Prime Numbers
Tip: Divisibility Rules
Use these shortcuts to check divisibility:
- By 2: Number ends in 0, 2, 4, 6, or 8
- By 3: Sum of digits is divisible by 3
- By 5: Number ends in 0 or 5
- By 7: Double the last digit, subtract from the rest
- By 11: Alternating sum of digits is divisible by 11