Find every factor, prime factorisation, and factor pair of any positive integer — with factor sum, prime check, and perfect-number check.
Results update live as you type.
Prime factorisation
Factor count
6
τ(n) — number of divisors
Sum of factors
168
σ(n) — divisor sum
Is prime?
No
has > 2 factors
Is perfect?
No
aliquot sum ≠ n
| Property | Definition | Example |
|---|---|---|
| Prime | Exactly 2 factors: 1 and itself | 7, 11, 97 |
| Composite | More than 2 factors | 4, 12, 60 |
| Perfect | Equals sum of proper divisors | 6, 28, 496 |
| Abundant | Aliquot sum > n | 12, 18, 24 |
| Deficient | Aliquot sum < n | 8, 9, 10 |
| Square | Odd number of factors | 4, 9, 36, 49 |
The Method
A factor of n is any positive integer that divides n exactly. The
straightforward algorithm tests every integer from 1 to √n: if i divides
n, both i and n / i are factors. Prime
factorisation repeatedly divides n by the smallest prime that fits,
until what remains is 1. By the fundamental theorem of arithmetic, every
integer greater than 1 has a unique prime factorisation (up to ordering).
Working for n = 60
About This Tool
A factor calculator — also called a divisor calculator or prime factorisation tool — finds every whole number that divides a given positive integer with no remainder. Enter any number and the calculator returns the full list of factors, the prime factorisation (with exponents), every factor pair whose product is your number, the sum and count of factors, and a check for whether the number is prime or perfect.
Factors are foundational throughout school and university mathematics. They underpin simplifying fractions, finding common denominators, factoring polynomials, integer factorisation in cryptography (the security of RSA depends on it being hard for very large numbers), and number-theoretic functions such as τ(n) (count of divisors), σ(n) (sum of divisors), and Euler's φ(n).
This calculator uses two complementary algorithms: a simple trial division up to √n to enumerate all factors (which is fast enough for any number you would compute by hand), and a repeated-division approach for the prime factorisation. Both run entirely in your browser and use IEEE 754 double-precision integers — accurate for any number up to 2⁵³ − 1 (about 9 × 10¹⁵). Beyond that range you need a dedicated big-integer library.
Use this free factor calculator for homework, exam revision, simplifying fractions and ratios, exploring perfect and abundant numbers, or as a teaching aid. All calculation is local — no sign-up, no tracking, no data sent to any server.
Every Factor
Complete factor list, count, and sum (τ and σ from number theory).
Prime Factorisation
Both expanded (2 × 2 × 3 × 5) and exponential (2² × 3 × 5) form.
Factor Pairs
All (a, b) such that a × b = n — useful for area, factoring, and divisibility.
Step-by-Step Working
Every division step of the prime factorisation, exactly as you'd write it.
100% Free & Private
No account, no tracking — every calculation runs locally in your browser.
Prime & Perfect Check
Instantly flags primes, perfect numbers, abundants and deficients.
One input gives you a full factor analysis in seconds.
Type any positive integer into the number field, or drag the slider to scan up to 1000. The calculator accepts values up to 100 million via direct input.
Every factor of n appears as a coloured pill, with primes highlighted. The aux label tells you exactly how many factors the number has — useful for spotting square numbers (always odd count).
Each factor pair (a × b = n) is shown beneath. Factor pairs are the answer set for "what two integers multiply to give n?" — essential in factoring quadratics and area / dimension problems.
The prime factorisation headlines the summary card in exponential form (e.g. 2² × 3 × 5). Use this to simplify fractions, find common denominators, and compute GCD / LCM with other numbers.
The bar chart shows the multiplicity of each prime factor (the exponent in p^k). Tall bars indicate the prime appears multiple times — useful for spotting square or higher-order powers in n.
The stat tiles flag prime / perfect / abundant / deficient, and the formula section shows the division steps exactly as you would write them out by hand.
Everything you need to know about factors, divisors, primes, and how to interpret your result.
A factor (or divisor) of a positive integer n is any positive integer that divides n exactly, with no remainder. The factors of 12 are 1, 2, 3, 4, 6, and 12 — six factors in total. Every positive integer has at least two factors: 1 and itself. Numbers with exactly these two factors are called prime; numbers with more are composite.
Prime factorisation expresses an integer as a product of prime numbers. For example 60 = 2 × 2 × 3 × 5, which is written more compactly in exponential form as 2² × 3 × 5. By the fundamental theorem of arithmetic, every integer greater than 1 has a unique prime factorisation (up to ordering). This makes the prime factorisation the canonical "fingerprint" of a number.
Factors divide a number — multiples are produced by multiplying a number. So 3 is a factor of 12 (because 3 divides 12), while 12 is a multiple of 3 (because 12 = 3 × 4). Every number is both a factor of itself and a multiple of itself. The factors of n are finite; the multiples of n are infinite (n, 2n, 3n, …).
A prime number has exactly two factors: 1 and itself. The first few primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. To check whether n is prime by hand, test divisibility by 2, by 3, then by odd numbers up to √n. If none divides n exactly, it is prime. Note that 1 is not prime by modern definition (a prime must have exactly two factors).
A perfect number equals the sum of its proper divisors (factors excluding itself). 6 is the smallest: 6 = 1 + 2 + 3. The next are 28, 496, and 8128. All known perfect numbers are even and follow Euclid's formula: 2^(p−1) × (2^p − 1) where 2^p − 1 is a Mersenne prime. Whether any odd perfect numbers exist is one of the oldest unsolved problems in mathematics.
If a number's aliquot sum (the sum of proper divisors) is greater than the number itself, it is abundant — the smallest is 12 (1 + 2 + 3 + 4 + 6 = 16 > 12). If the aliquot sum is less than the number, it is deficient (8: 1 + 2 + 4 = 7 < 8). And if the aliquot sum exactly equals the number, it is perfect. Every prime number is deficient (its aliquot sum is just 1).
Factor pairs (a, b) where a × b = n appear naturally in problems involving area and perimeter (a rectangle with area 60 has integer-sided shapes for each pair: 1×60, 2×30, 3×20, 4×15, 5×12, 6×10), simplifying fractions, factoring quadratics, and finding integer solutions to equations. For a square number, one of the pairs is (√n, √n).
Factors usually pair up: if a divides n then so does n/a, and a ≠ n/a unless a = √n. For a square number, the square root is its own pair, so it is counted only once. That gives squares an odd factor count: 36 has 9 factors (1, 2, 3, 4, 6, 9, 12, 18, 36), with 6 being its own pair. This is why squares are exactly the integers with an odd τ(n).
The calculator uses IEEE 754 double-precision arithmetic and is exact for integers up to 2⁵³ − 1 ≈ 9 × 10¹⁵ (about 9 quadrillion). The factoring algorithm itself is trial division up to √n, which is fast for any number you would input by hand. For genuinely huge numbers (RSA-scale, 100+ digits) you need a dedicated big-integer library and far more sophisticated factoring algorithms (Pollard rho, the quadratic sieve, the general number field sieve).
These are standard number-theoretic functions: τ(n) (sometimes written d(n)) is the number of divisors of n; σ(n) is the sum of divisors. For n = 60: τ(60) = 6 and σ(60) = 1 + 2 + 3 + 4 + 5 + 6 + 10 + 12 + 15 + 20 + 30 + 60 = 168. (Note: σ includes n itself; the aliquot sum s(n) is σ(n) − n.)
1 is not prime by definition: a prime must have exactly two positive factors, and 1 has only one. Excluding 1 keeps the fundamental theorem of arithmetic clean — every integer greater than 1 has a unique prime factorisation. If 1 were prime, factorisations would no longer be unique (since you could multiply by any number of 1s).
Integer factorisation is the foundation of RSA encryption: a public RSA key is a product of two large secret primes. Security depends on the assumption that factoring a several-hundred-digit number is computationally infeasible with current algorithms and hardware. The best general-purpose factoring algorithm (the general number field sieve) is sub-exponential but still wildly impractical at RSA-2048 sizes — which is why RSA remains secure for now.