GCF of monomials calculator
Get the greatest common factor of algebraic terms, variables included: 12x²y, 18xy³ and 24x³y share 6xy. Only letters present in every term are kept, each at its lowest power. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Terms
- 12x^2y, 18xy^3, 24x^3y
- Gcf
- 6xy
- Coefficient
- 6
Finding the greatest common factor of monomials is the first move in factoring polynomials by grouping, pulling a common factor out of an expression, and simplifying ratios of algebraic terms. Students, tutors, and automation scripts need a rule that never forgets a variable power and never invents a factor that is not shared by every term. This greatest common factor of monomials calculator accepts two or more monomial strings, takes the greatest common divisor of their absolute integer coefficients, keeps only the variables that appear in every term, raises each shared variable to the minimum exponent present across the list, and returns the product as a single normalized GCF string.
What the GCF of monomials means and how the algorithm works
A monomial is a single term built from an optional signed integer coefficient and single-letter variables each raised to a non-negative integer power, for example twelve x squared y, negative eight a cubed b, or a pure number such as forty-two. The greatest common factor of a list of monomials is the largest monomial that divides every term in the list without remainder. Algebra textbooks build that factor in two independent pieces.
First take the greatest common divisor of the absolute values of the coefficients using the Euclidean algorithm; the coefficient of the GCF is conventionally non-negative. Second, for every variable letter, look at the exponent of that letter in each monomial, treat a missing letter as exponent zero, and keep the minimum of those exponents. Only variables whose minimum is strictly positive appear in the GCF.
Multiplying the coefficient GCD by each shared variable raised to its minimum exponent yields the common factor. Example: the terms twelve x squared y, eighteen x y cubed, and twenty-four x cubed y share coefficient gcd six, share x to the first power and y to the first power, and therefore have GCF six x y. Terms that share no variables still share a numeric GCF when their coefficients have a common divisor greater than one.
Pure powers such as x squared and x cubed produce a GCF of x squared with coefficient one.
What to enter and what you get back
Each string may include an optional leading plus or minus, an integer coefficient, and single-letter variables with optional caret exponents; spaces and asterisk multiplication signs are stripped before parsing. Shorthand forms such as x2 are accepted as x to the power two. Decimal or fractional coefficients are rejected because classic school GCF of monomials works over the integers.
On success the body echoes the terms as normalized strings, a parsed array of coefficient and variable maps for inspection, the gcf display string, the non-negative coefficient of the GCF, a variables object mapping each shared letter to its minimum exponent, and a short formula string restating the two-piece rule. Example: terms twelve x squared y, eighteen x y cubed, twenty-four x cubed y returns gcf six x y with coefficient six and variables x one, y one.
Where a monomial GCF calculator earns its keep
Classrooms and tutoring platforms use a greatest common factor of monomials calculator so learners can check the common factor before they factor a full polynomial by grouping or by the distributive property. Exam-prep generators need a deterministic oracle that marks wrong a GCF that drops a shared variable or inflates an exponent beyond the true minimum. Failed validation does not consume balance.
Keep term lists small, pin the normalized gcf string in tests, and you get reproducible common factors for homework, LMS plugins, and edge workers without symbolic math infrastructure of your own.
What you can do with it
Classroom and homework checks
Let students submit two or more monomials and verify the GCF coefficient and shared variable powers against textbook factoring methods.
Exam and LMS auto-grading
Mark GCF-of-monomials items with a deterministic oracle that rejects invalid syntax and never invents factors missing from any term.
Notebooks and symbolic cleanup
Pull a common monomial factor from lists of algebraic terms in scripts without importing a full computer-algebra library.
FAQ
What monomial syntax is accepted?
Optional signed integer coefficient, single-letter variables, optional ^exponents (or trailing digits like x2). Spaces and * are ignored. Decimals and multi-letter variable names are rejected.
What if the terms share no variables?
The GCF is then only the numeric gcd of the coefficients (possibly 1). Shared variables appear only when every term has that letter with a positive minimum exponent.
How do I find the GCF of 12x²y, 18xy³ and 24x³y?
Take the greatest common divisor of 12, 18 and 24, which is 6. Then keep x and y, each at the lowest power found in the three terms. The result is 6xy.
Why is the lowest exponent used for each variable?
The factor has to divide every term. A higher power of x would not divide the term that contains x only once, so the smallest exponent is the largest that fits all of them.
Can the greatest common factor be negative?
No. The number part is worked out from the absolute values of the coefficients, so it is always non-negative, even when some of your terms carry a minus sign.
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/algebra/gcf-of-terms \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"terms":["12x^2y","18xy^3","24x^3y"]}'const res = await fetch("https://api.kit.forhosting.com/algebra/gcf-of-terms", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"terms": [
"12x^2y",
"18xy^3",
"24x^3y"
]
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/algebra/gcf-of-terms",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"terms": [
"12x^2y",
"18xy^3",
"24x^3y"
]
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/algebra/gcf-of-terms", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"terms":["12x^2y","18xy^3","24x^3y"]}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"terms":["12x^2y","18xy^3","24x^3y"]}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/algebra/gcf-of-terms", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"terms": [
"12x^2y",
"18xy^3",
"24x^3y"
]
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "algebra.gcf_of_terms",
"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
Failed validation is not billed; successful calls use the published $0.002 base price with no model tokens and no outbound network.
What the GCF of monomials means and how the algorithm works
The endpoint never factors across addition or subtraction inside a term, because a true monomial has no plus or minus between variable products; such input is rejected as invalid rather than partially interpreted.
Input contract, response shape, and error behaviour
POST a JSON body with a required array field named terms containing between two and twenty monomial strings. Errors use invalid_input and are not billed: fewer than two terms, more than twenty, empty strings, non-string items, invalid characters, lone signs, unsafe integer coefficients, and the pathological case where every coefficient is zero. Pair the structured variables map with your own pretty-printer if you need superscripts in HTML; this endpoint returns plain ASCII caret notation for stable goldens and SDK clients.
Where a monomial GCF calculator earns its keep
Spreadsheet and notebook workflows often pull a common monomial out of rows of symbolic terms when reducing physics formulas or engineering expressions; calling a tiny parse API avoids standing up a full computer-algebra system for coefficient GCD plus min exponents.
Technical questions
How do I find the GCF of two or more monomials?
Send a terms array of monomial strings. The API returns gcd of absolute coefficients times each shared variable raised to the minimum exponent across all terms.
How many monomials can I send?
Between 2 and 20 inclusive. Fewer than two or more than twenty fails with invalid_input and is not billed.
What does the GCF of monomials calculator cost?
The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.