Factors Calculator — Find All Factors, Factor Pairs & Prime Factorization
Find all factors of any number, see factor pairs, and get the full prime factorization tree. Understand the connection to GCF and LCM with worked examples.
A factor of a number n is any integer that divides n exactly, leaving no remainder. Finding all factors is fundamental to simplifying fractions, finding GCF and LCM, and understanding divisibility. List every factor instantly with the CalcHub Factors Calculator.
How to Find All Factors
Method: Test every integer from 1 to √n. If i divides n evenly, both i and n/i are factors. Find all factors of 48: √48 ≈ 6.93, so test 1 through 6:| Divisor | Quotient | Factor pair |
|---|---|---|
| 1 | 48 | (1, 48) |
| 2 | 24 | (2, 24) |
| 3 | 16 | (3, 16) |
| 4 | 12 | (4, 12) |
| 5 | 48/5 = 9.6 | Not a factor |
| 6 | 8 | (6, 8) |
Prime Factorization
Breaking a number into its prime building blocks.
Factor tree for 360:360
├── 2 × 180
├── 2 × 90
├── 2 × 45
├── 3 × 15
├── 3 × 5For 360 = 2³ × 3² × 5¹:
Number of factors = (3+1)(2+1)(1+1) = 4 × 3 × 2 = 24 factors
Factors of Common Numbers
| Number | Factors | Count |
|---|---|---|
| 12 | 1, 2, 3, 4, 6, 12 | 6 |
| 24 | 1, 2, 3, 4, 6, 8, 12, 24 | 8 |
| 36 | 1, 2, 3, 4, 6, 9, 12, 18, 36 | 9 |
| 48 | 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 | 10 |
| 60 | 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 | 12 |
| 100 | 1, 2, 4, 5, 10, 20, 25, 50, 100 | 9 |
GCF and LCM from Prime Factorization
Greatest Common Factor (GCF): Use the smallest exponent for each shared prime.GCF(48, 36):
48 = 2⁴ × 3
36 = 2² × 3²
GCF = 2² × 3 = 12
LCM(48, 36):
LCM = 2⁴ × 3² = 16 × 9 = 144
Perfect Squares and Their Factor Counts
Perfect squares always have an odd number of factors. That's because for every factor pair (a, b) with a ≠ b, both are counted separately — but when a = b = √n, it's counted once. The square root "pairs with itself."
36 = 6² has factors 1, 2, 3, 4, 6, 9, 12, 18, 36 → 9 factors (odd) ✓
What's the difference between a factor and a multiple?
Factors divide into n; multiples n divides into. The factors of 12 are 1, 2, 3, 4, 6, 12. The multiples of 12 are 12, 24, 36, 48, ... They go in opposite directions — factors go smaller, multiples go larger.
How do highly composite numbers work?
A highly composite number has more factors than any smaller positive integer. The sequence starts: 1, 2, 4, 6, 12, 24, 36, 48, 60, 120... Many calendar and clock systems are based on 12 and 60 because they have so many factors (6 and 12 respectively), making division into equal parts easy.
What is a perfect square factor?
A factor that is itself a perfect square. For √72: since 72 = 36 × 2 = 6² × 2, the perfect square factor is 36, and √72 = 6√2. Finding the largest perfect square factor is how you simplify square roots without a calculator.