Factor Calculator
Calculate all factors and prime factors of any number.
What are Factors?
Factors are numbers that divide evenly into another number. For example, the factors of 12 are: 1, 2, 3, 4, 6, and 12 (since each divides 12 with no remainder).
Prime Factorization is the process of breaking down a number into its prime factors (factors that are only divisible by 1 and themselves). For example, 12 = 2 × 2 × 3.
Understanding factors is important in mathematics, especially in working with fractions, finding common denominators, and solving number puzzles.
Learning Guide
Understanding Factors vs Multiples
Factors are numbers that divide evenly into a given number. Multiples are numbers that result from multiplying a given number by other whole numbers.
For the number 6:
Factors of 6: 1, 2, 3, 6 (numbers that divide 6 evenly)
Multiples of 6: 6, 12, 18, 24, 30... (6×1, 6×2, 6×3...)
Think of it this way: factors are what goes into a number, while multiples are what comes out of multiplying that number.
Prime Factorization & Why It Matters
Prime factorization expresses a number as a product of prime numbers only. Every number greater than 1 can be uniquely represented this way—known as the Fundamental Theorem of Arithmetic.
Example: Breaking down 60
60 = 2² × 3 × 5 (or written out: 60 = 2 × 2 × 3 × 5)
Why it matters:
- • Finding GCD (Greatest Common Divisor) and LCM (Least Common Multiple)
- • Simplifying fractions and working with ratios
- • Cryptography and computer security
- • Understanding number patterns and properties
Finding GCD Using Prime Factorization
The Greatest Common Divisor (GCD) is the largest number that divides both numbers evenly. Using prime factorization, multiply only the common prime factors with the lowest power.
Example: Find GCD(48, 36)
48 = 2⁴ × 3¹
36 = 2² × 3²
Common factors: 2² and 3¹
GCD = 2² × 3 = 12
Prime Numbers vs Composite Numbers
Prime Numbers
Exactly two factors: 1 and themselves
2, 3, 5, 7, 11, 13, 17, 19, 23, 29...
Composite Numbers
More than two factors
4, 6, 8, 9, 10, 12, 14, 15...
Special notes:
• The number 1 is neither prime nor composite
• The number 2 is the only even prime number
Quick Divisibility Rules
Check divisibility without long division using these shortcuts:
÷ 2: Last digit is even (14, 26, 38)
÷ 3: Sum of digits divisible by 3 (276 → 2+7+6=15 → ✓)
÷ 5: Last digit is 0 or 5 (15, 20, 35)
÷ 9: Sum of digits divisible by 9 (387 → 3+8+7=18 → ✓)
÷ 10: Last digit is 0 (10, 50, 120)
Real-Life Applications
Bus Scheduling
Bus A arrives every 6 minutes, Bus B every 8 minutes. LCM(6,8) = 24, so both arrive together every 24 minutes.
Land Division
A 48m × 36m field needs equal square plots. GCD(48,36) = 12m, so each square is 12m × 12m.
Recipe Scaling
A recipe serves 12 people but you need to serve 18. Use GCD to scale proportionally without waste.
Prime Factorization & Fractions
Simplifying Fractions: 72/120
72 = 2³ × 3², and 120 = 2³ × 3 × 5. Cancel common factors: 72/120 = 3/5
Finding LCD: 1/6 + 1/8
6 = 2 × 3 and 8 = 2³. LCD = 24, so: 1/6 + 1/8 = 4/24 + 3/24 = 7/24
Perfect Squares & Perfect Cubes
A perfect square has all even exponents in its prime factorization. A perfect cube has exponents divisible by 3.
Perfect Squares
64 = 2⁶ = (2³)² = 8²
√64 = 8
Perfect Cubes
64 = 2⁶ = (2²)³ = 4³
∛64 = 4
How to Use This Calculator for Learning
Enter any number up to 10 trillion
Start with 12, 24, or 60 to understand the concepts
Click Calculate for instant results
See all factors, pairs, and factorization tree
Verify your homework
Use the tree to understand step-by-step breakdown
Learn with real examples
Explore the sections above for patterns