ForHosting KIT · Developer Utilities

Is modulo associative?

The modulo associativity API answers a deceptively simple question: for three given integers, does grouping matter?

● BetaFree · in your browser

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It computes both orderings of the remainder operation — (a % b) % c on the left and a % (b % c) on the right — and tells you whether they agree. You send three integers and get back both sides of the comparison plus a plain true or false, computed over arbitrary-precision integers so the answer is exact for values of any realistic size. If any divisor position would be zero — b, c, or the intermediate remainder b % c — the request is rejected with a clear error instead of a division-by-zero surprise. It is a small check, but it is the one that decides whether a chained-modulo expression in your code means what you think it means.

Why grouping matters for the remainder operation

Addition and multiplication are associative: (1 + 2) + 3 equals 1 + (2 + 3), always, so nobody thinks about parentheses. The remainder operation enjoys no such guarantee. The expression a % b % c is ambiguous on paper, and most programming languages resolve it by evaluating left to right, producing (a % b) % c. Whether that equals the alternative grouping a % (b % c) depends entirely on the three values involved. Take a = 17, b = 5, c = 3: the left grouping yields (17 % 5) % 3 = 2 % 3 = 2, while the right grouping yields 17 % (5 % 3) = 17 % 2 = 1. The two groupings disagree, so modulo is not associative for that triple. But for a = 8, b = 5, c = 3 both sides come out to 0, and the operation happens to be associative there. This capability exists because that distinction is easy to get wrong in a code review and tedious to verify by hand across a whole set of candidate parameters: one call settles it deterministically, returning both intermediate values so you can see exactly where the two groupings diverge instead of taking a bare boolean on faith.

How the check is computed, and the zero-divisor rule

The computation is deliberately transparent. First each input is parsed as an integer — decimal strings or JSON numbers are both accepted, and arbitrary-precision arithmetic is used internally so there is no precision ceiling at 2⁵³ and no rounding anywhere in the pipeline. Then the left side is evaluated as (a % b) % c and the right side as a % (b % c), where % follows the truncated convention familiar from JavaScript, C and Java: the result takes the sign of the dividend. The two results are compared for exact equality, and the answer is reported alongside both sides. There is one situation where no boolean answer exists: division by zero. Three divisor positions appear in the two expressions — b in a % b, c in (a % b) % c, and the intermediate value b % c as the divisor on the right-hand side. If b or c is zero, or if b % c happens to be zero for the given values, the request is rejected as invalid input with a message naming exactly which divisor failed. That error is never billed, so you can safely probe parameter ranges programmatically. Inputs up to 2048 decimal digits per field are accepted.

Where this check earns its place

The most common caller is someone teaching or learning number theory: associativity questions appear constantly in textbooks and problem sets, and being able to verify a claim over hundreds of triples in seconds turns a chore into an experiment. The second caller is a developer refactoring hash-bucket or sharding logic, where expressions like h % m % n sneak in and silently change meaning depending on evaluation order — checking associativity over the actual modulus pairs in production tells you whether a rewrite is safe. The third is anyone building a symbolic math or expression-simplification tool, which must know which rewrite rules are valid before applying them. The free calculator on this page runs the exact same code as the paid endpoint, so what you test in the browser is what your integration will receive. Automated calls cost $0.002 per request with no per-unit surcharge. Send a JSON object with fields a, b and c; the response echoes the inputs, reports left and right as decimal strings, and gives the final verdict in the associative field — true when both groupings agree, false when they do not.

What you can do with it

Verify a number-theory exercise set

Check associativity claims over dozens of triples in seconds instead of hand-computing each grouping, and see both sides when they disagree.

Audit hash-bucket and sharding expressions

Confirm whether h % m % n can be safely regrouped in production code before refactoring an expression that routes traffic.

Power an expression-simplification rule

Let a symbolic math tool decide whether reassociating a chained modulo is valid for concrete values before applying the rewrite.

FAQ

What does it cost?

$0.002 per request with no per-unit surcharge. It is also free to run in your browser on this page.

What exactly does it compare?

The left grouping (a % b) % c against the right grouping a % (b % c). It returns both values and true when they are equal.

What happens if a divisor is zero?

The request is rejected as invalid input. That covers b = 0, c = 0, and the subtler case where b % c = 0, which would make the right-hand divisor zero.

Can the integers be negative?

Yes. The remainder follows the truncated convention of JavaScript, C and Java: the result takes the sign of the dividend.

How large can the integers be?

Up to 2048 decimal digits per field, computed with arbitrary-precision arithmetic, so there is no rounding or 2⁵³ ceiling.

Is the answer cached or random?

No. The computation is fully deterministic: the same three integers always produce the same left, right and verdict.

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

POSThttps://api.kit.forhosting.com/math/is-modulo-associative

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/math/is-modulo-associative \
  -H "Authorization: Bearer $KIT_KEY" \
  -H "Content-Type: application/json" \
  -d '{"a":"17","b":"5","c":"3"}'

const res = await fetch("https://api.kit.forhosting.com/math/is-modulo-associative", {
  method: "POST",
  headers: {
    "Authorization": `Bearer ${process.env.KIT_KEY}`,
    "Content-Type": "application/json"
  },
  body: JSON.stringify({
    "a": "17",
    "b": "5",
    "c": "3"
  })
});
const { task_id } = await res.json();

import os, requests

res = requests.post(
    "https://api.kit.forhosting.com/math/is-modulo-associative",
    headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
    json={
    "a": "17",
    "b": "5",
    "c": "3"
},
)
task_id = res.json()["task_id"]

<?php
$res = file_get_contents("https://api.kit.forhosting.com/math/is-modulo-associative", false, stream_context_create([
  "http" => [
    "method"  => "POST",
    "header"  => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
    "content" => '{"a":"17","b":"5","c":"3"}',
  ],
]));
$task = json_decode($res, true);

body := bytes.NewBufferString(`{"a":"17","b":"5","c":"3"}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/math/is-modulo-associative", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)

Example request

{
  "a": "17",
  "b": "5",
  "c": "3"
}

Example response

{
  "task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
  "type": "math.is_modulo_associative",
  "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

Per request$0.002

Published price — no tokens, no invented credits. A failed task is never charged.

Limits

max_digits 2048

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.

Read the full KIT documentation →