Pick your mode
FACTOR decomposes n into primes, DIVISORS lists every factor with σ(n) and the aliquot class, CHECK delivers a primality verdict. Three tabs, top of the instrument.
FACTOR · DIVISORS · CHECK
01
The instrument — stripped to its primes
The engine trial-divides any number to the bone — the prime factorization with every division written out, the complete divisor dossier (τ, σ, aliquot class), Euler's totient, and a deterministic primality check. No equals button; it re-solves on every keystroke.
NUMBER · n
WHOLE NUMBER · 2 – 10¹² · COMMAS & SPACES OK · SOLVES ON EVERY KEYSTROKE
TRY
TRIAL DIVISION — THE WORKLIVE
— NO WORK YET · TYPE A NUMBER ABOVE —
TRIAL-DIVISION CORE·
RANGE 2 – 10¹²·
INTEGER-EXACT·
NO NETWORK CALLS
02
How to use it
Three modes, one input box, no equals button — the whole flow takes under a minute to learn and then it is muscle memory.
Enter any whole number
From 2 up to 10¹² — one trillion. Commas and spaces are stripped automatically, and the arithmetic stays exact integer the whole way through.
n = 1,234,567 → parses clean
No equals button
The screen re-solves on every keystroke — change one digit and the entire proof panel rewrites itself instantly.
live · every keystroke
Read the dossier
Exponent form up top; beneath it ω, Ω, τ, σ, φ, the aliquot class and the largest prime factor — every row a textbook asks for.
τ · σ · φ — all present
Follow the work
Every division is written out, rejected candidates are tallied, and a product check closes the proof. Copy the lines straight into your notes.
360 ÷ 2 = 180 … line by line
Tap the example chips
Perfect numbers, the 91 = 7 × 13 trap, and primes that survive to the last candidate — one tap each, pre-loaded and solved.
28 · 91 · 496 · 8,128
factorize/360/trial division · √bound ≈ 18.97
n = 360
÷ 2180
÷ 290
÷ 245
÷ 315
÷ 35
5 — no divisor ≤ √5PRIME ✓
360 = 23 × 32 × 5
τ 24 · σ 1,170 · φ 96
03
Questions, factored
Prime factors, divisors, σ, φ and primality — the questions that come up in class and in the search bar, answered properly.
Q1What is prime factorization?
Writing a number as a product of primes only. 360 = 2 × 2 × 2 × 3 × 3 × 5, usually tidied to 2³ × 3² × 5. Every whole number from 2 up has exactly one such decomposition apart from order — that guarantee is the Fundamental Theorem of Arithmetic, and it is why the calculator can promise THE answer, not AN answer.
Q2How do I find the prime factors of a number by hand?
Trial division. Divide by 2 as long as it goes in, recording each quotient; then move to 3, then 5, 7, 11 and so on. When the candidate squared is bigger than what remains, you are done — the leftover, if greater than 1, is itself prime. The FACTOR tab writes out every one of these lines so you can copy the working straight into homework.
Q3Why only test divisors up to √n?
If n = a × b, the smaller factor is at most √n — two numbers both bigger than √n multiply to more than n. So if nothing up to √n divides, nothing at all does, and n is prime. The bound is what keeps hand factoring feasible: 997 needs testing only up to 31, not to 996.
Q4How do I list all divisors from the prime factorization?
Raise each prime to every power from 0 up to its exponent and multiply — the count is the product of (exponent + 1). For 360 = 2³ × 3² × 5: (3+1)(2+1)(1+1) = 24 divisors, generated as 2ᵃ·3ᵇ·5ᶜ. The DIVISORS tab lists them sorted, with the count, sum and aliquot class attached.
Q5What is the difference between a prime factor and a divisor?
The divisors of 12 are 1, 2, 3, 4, 6 and 12 — six of them, mostly composite. Its prime factors are just 2 and 3 (with repetition, 2 × 2 × 3). Prime factors are the atoms; divisors are every number you can build from those atoms, plus 1.
Q6What are perfect, abundant and deficient numbers?
Compare the aliquot sum s(n) = σ(n) − n — the sum of the proper divisors — with n itself. Equal means perfect, like 6 (1+2+3) and 28 (1+2+4+7+14). Greater means abundant, like 12. Less means deficient, like 8. Tap 28, 496 or 8,128 in DIVISORS mode to watch a perfect number prove itself.
Q7How does the primality check work?
Deterministic trial division with the 6k ± 1 wheel: after 2 and 3, every prime is one more or one less than a multiple of 6, so only those candidates are tested, up to ⌊√n⌋. For the tool's range of 2 to 10¹² this is exact — no probabilistic shortcuts. 91 is the classic trap: it looks prime, but 7 × 13 says otherwise.
Q8What are ω(n), Ω(n), τ(n), σ(n) and φ(n)?
The number theorist's dashboard. ω(n) counts the distinct prime factors, Ω(n) counts them with multiplicity, τ(n) counts all divisors, σ(n) sums all divisors, and Euler's φ(n) counts how many numbers below n share no factor with n. The rows under the calculator screen report these for whatever you type.
Q9Is it free, and does my number leave the page?
Completely free — no sign-up, no locked features. Every division runs locally in your browser; nothing is recorded or transmitted, and the page keeps working offline once it has loaded.