Factors Calculator
Find all factors (divisors) of any number instantly. See factor pairs, prime factorization, and whether a number is prime.
About the Factors Calculator
A factor (or divisor) of a whole number n is any integer that divides n exactly with no remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12 — because each of those numbers divides 12 without leaving a remainder. Every positive integer has at least two factors: 1 and itself. Numbers that have exactly two factors (1 and themselves) are called prime numbers. Numbers with more than two factors are called composite numbers.
Finding the factors of a number is a fundamental operation in number theory with practical applications in algebra, fractions, cryptography, and puzzle solving. When simplifying fractions, you need the greatest common factor (GCF) of the numerator and denominator. When finding the least common multiple (LCM) of two numbers, you often use prime factorization as an intermediate step. When solving quadratic equations by factoring (the diamond problem), you need factors of the constant term that add up to the coefficient of the middle term.
Prime factorization expresses any composite number as a unique product of prime numbers. For example, 360 = 2³ × 3² × 5 = 8 × 9 × 5. This is guaranteed by the Fundamental Theorem of Arithmetic, which states that every integer greater than 1 has a unique prime factorization (ignoring the order of factors). Our calculator finds all factors of any number up to 1,000,000 instantly, displays them color-coded to distinguish primes from composites, shows the prime factorization with exponent notation, and lists all factor pairs.
Pros & Cons
- +Finds all factors, prime factors, and factor pairs in one step
- +Color-codes prime factors in green for quick identification
- +Shows prime factorization in standard exponent notation (e.g. 2³ × 3²)
- +Indicates whether the input number itself is prime
- +Works up to 1,000,000 instantly with no loading delay
- −Limited to 1,000,000 — larger numbers require more powerful tools
- −Does not find the GCF or LCM of two numbers (use a dedicated GCF/LCM calculator)
- −Does not work with decimals or negative numbers
- −Prime testing above 1,000,000 requires a more efficient primality test
- −Does not show step-by-step factor tree diagrams
How to Find Factors Step by Step
To find all factors of a number n manually, test every integer from 1 up to the square root of n. For each integer i that divides n evenly (n mod i = 0), both i and n/i are factors. This pair-finding approach is efficient because factors come in pairs. For 36: √36 = 6, so test 1 through 6. 36/1=36 ✓ (pair: 1,36), 36/2=18 ✓ (pair: 2,18), 36/3=12 ✓ (pair: 3,12), 36/4=9 ✓ (pair: 4,9), 36/5=7.2 ✗, 36/6=6 ✓ (pair: 6,6 — only counted once). Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36.
For prime factorization, use trial division: divide by 2 repeatedly until odd, then try 3, 5, 7, 11, and so on through all primes up to √n. For 360: divide by 2 → 360/2=180, 180/2=90, 90/2=45 (can't divide by 2 anymore). Divide by 3 → 45/3=15, 15/3=5 (can't divide by 3). 5 is prime. Result: 2³ × 3² × 5. The total number of factors equals (3+1)(2+1)(1+1) = 24 — multiply one more than each exponent in the prime factorization.
The Fundamental Theorem of Arithmetic guarantees that this prime factorization is unique. Two numbers that share prime factors — like 12 = 2² × 3 and 18 = 2 × 3² — have a GCF of 2 × 3 = 6 (the minimum of each exponent) and an LCM of 2² × 3² = 36 (the maximum of each exponent). This relationship GCF(a,b) × LCM(a,b) = a × b is a powerful identity used throughout number theory.