Powerful part calculator
Splits any positive whole number into its largest powerful divisor and a squarefree cofactor: 12 becomes 4 × 3, because 2 appears squared and 3 only once. The two pieces always multiply back to n and share no prime. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- N
- 12
- Powerful part
- 4
- Squarefree cofactor
- 3
Every positive integer factors uniquely into primes raised to powers, and those powers fall into two clean buckets. Exponents of two or more build a powerful number — if a prime divides that part, its square does too. Exponents equal to one build a square-free cofactor, a product of distinct primes with no squared factor. This powerful part calculator takes a single positive integer n, peels it by trial division, and returns the largest powerful divisor together with the complementary squarefree cofactor so that n equals their product with gcd one.
What powerful part and squarefree cofactor mean
A positive integer m is called powerful (also squarefull) when every prime that divides m also divides m to the second power: if p divides m then p squared divides m. Equivalently, in the prime factorization of m every exponent is at least two, so m can never be divisible by a prime to the first power only. Classic examples are one (vacuously), four, eight, nine, twelve is not powerful because of the lone factor three, while thirty-six equals two squared times three squared and is powerful.
A positive integer is square-free when it is not divisible by any perfect square other than one, which means every exponent in its factorization is zero or one — products of distinct primes such as six, ten, fifteen, or a single prime itself. Given any n greater than or equal to one, write n as a product of prime powers. Collect every power whose exponent is at least two into the powerful part; collect every prime that appears with exponent exactly one into the squarefree cofactor.
The product of those two pieces recovers n, they are always coprime, and the powerful part is precisely the largest powerful divisor of n. When n itself is powerful the squarefree cofactor is one; when n is square-free the powerful part is one; when n is one both pieces are one. This decomposition is unique and is the standard way textbooks and OEIS sequences separate the squarefull core of an integer from the primes that only appear once.
Where the powerful–squarefree split earns its keep
Classroom homework and contest programming problems often ask whether an integer is powerful, square-free, or neither, or they need the largest squarefull divisor before applying Möbius, radical, or abundance formulas. Cryptography demos and number-theory notebooks use the split to illustrate that every n is a product of a squarefull kernel and a square-free shell with gcd one, which is a stepping stone to understanding square-free moduli and powerful orders in multiplicative groups.
Content sites that rank for powerful part calculator or squarefree cofactor benefit from a live tool whose free browser path matches the paid automation path byte-for-byte on the same pure module. Failed validation does not consume balance.
What you can do with it
Homework and contest verification
Check student or contest answers for the largest powerful divisor and the complementary squarefree cofactor of any positive integer.
FAQ
Is the powerful part calculator free?
The browser tool runs free on your device.
Why must n be at least 1?
The powerful–squarefree decomposition is defined for positive integers.
When is the powerful part equal to n itself?
Exactly when n is powerful: every prime exponent is at least two (or n is 1).
What is the powerful part of 360?
It is 72, with a squarefree cofactor of 5. 360 factors as 2 cubed times 3 squared times 5, so the cube and the square go to the powerful part and the lone 5 stays outside.
What does a prime number give?
A prime such as 17 has powerful part 1 and squarefree cofactor 17, because its only prime appears just once. The same happens with any product of distinct primes.
Related tools
For developers — API access
Everything on this page is available programmatically. This section is for teams who want to wire it into their own systems; everyone else can just use the tool above.
API endpoint
Prefer to automate it? One authenticated POST creates the task; the result comes back by webhook or a signed link. The same capability also runs here on the web, by email and from Telegram — and soon from our app too.
Call it from your stack
curl -X POST https://api.kit.forhosting.com/numth/powerful-part \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"n":12}'const res = await fetch("https://api.kit.forhosting.com/numth/powerful-part", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"n": 12
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/numth/powerful-part",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"n": 12
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/numth/powerful-part", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"n":12}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"n":12}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/numth/powerful-part", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"n": 12
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "numth.powerful_part",
"status": "queued",
"_links": {
"result": "/tasks/tsk_…/result"
}
}The API is asynchronous: the call returns a task_id immediately and the result arrives by webhook. Polling is capped at 1 req/s per task.
Pricing
Published price — no tokens, no invented credits. A failed task is never charged.
Errors
| HTTP | Code | Meaning |
|---|---|---|
401 | unauthorized | Missing or invalid API key. |
402 | insufficient_balance | Your balance doesn't cover the task price. |
404 | unknown_type | That task type doesn't exist. |
429 | rate_limited | Too many requests. Use the webhook instead of polling. |
Technical notes
Students use it to check homework on squarefull numbers; contest programmers feed the structured split into radical, abundance, or Möbius pipelines; and notebooks get a deterministic JSON contract free in the browser or at $0.002 per successful API call, with invalid input never billed.
How to call the calculator and read the result
Internally the engine validates the integer, factors it by peeling off twos and then walking odd trial divisors up to the square root of the remainder, and splits each prime power according to its exponent. Example: n equals twelve factors as two squared times three, so the powerful part is four and the squarefree cofactor is three. Example: n equals three hundred sixty factors as two cubed times three squared times five, so the powerful part is seventy-two and the squarefree cofactor is five. Example: a prime such as seventeen yields powerful part one and squarefree cofactor seventeen. Submit a JSON object with a single required field n — a safe JavaScript integer or a plain decimal integer string without commas, spaces, scientific notation, or leading zeros beyond the number zero itself. Values must be integers at least one; zero, negatives, fractions, Infinity, and NaN fail with invalid_input and are never charged. Alias field names number and value are accepted for convenience. The success payload echoes n, reports powerful_part and squarefree_cofactor as integers, sets is_powerful true exactly when the squarefree cofactor is one, sets is_squarefree true exactly when the powerful part is one, lists the ordered factors array of base and exponent pairs, and includes a short formula string restating the split rule. Stay within the safe integer range so every product stays exact in IEEE double integers; beyond that bound the request fails fast rather than inventing rounded factors.
Integration use cases
Number-theory pipelines
Emit powerful_part and squarefree_cofactor as structured integers for radical, Möbius, abundance, or squarefree-modulus helpers.
Classroom demos of powerful numbers
Show that n factors as a squarefull core times a product of distinct primes with a live deterministic widget and matching API.
Technical questions
How do I split an integer into its powerful part and squarefree cofactor?
POST an object with field n set to an integer ≥ 1. The response returns powerful_part, squarefree_cofactor, is_powerful, is_squarefree, the ordered factors list, and a formula string.
Is the result deterministic across browser and API?
Yes. The same pure trial-division routine powers both paths, so identical inputs produce identical JSON with no network or randomness involved.