# Book Isle Euclidean algorithm: trace divisors, remainders and the stopping condition

Language: en
Canonical: [Book Isle Euclidean algorithm: trace divisors, remainders and the stopping condition](<https://allluckly.cn/en/guides/nasbook-euclidean-gcd-remainder-chain-lcm/>)
中文：[中文 HTML](<https://allluckly.cn/guides/nasbook-euclidean-gcd-remainder-chain-lcm/>)
English: [English HTML](<https://allluckly.cn/en/guides/nasbook-euclidean-gcd-remainder-chain-lcm/>)
Published: 2026-11-12T00:00:00+08:00
Modified: 2026-11-12T09:43:29+08:00
Author: Bison

Trace forty-eight and eighteen while separating the last nonzero remainder, the zero stop and the least common multiple.

English translation of the [Chinese original](<https://allluckly.cn/guides/nasbook-euclidean-gcd-remainder-chain-lcm/>), translated on 2026-11-12. Original publication dates, attribution, code and version notes are retained.

A remainder chain ending in zero does not make the greatest common divisor zero. Read an original positive-integer example in Book Isle, map every division to the next input pair and verify common factors manually. External records are allowed; no algorithm executor or automatic proof is assumed, and GCD remains distinct from LCM.

## 1. Fix integer inputs and the algorithm scope

Prepare an original passage with positive integer inputs forty-eight and eighteen. Record file, edition, chapter/page and a nearby phrase, labeling the current dividend and divisor rather than fixing roles by screen order. This is not approximate decimal division or measurement precision. Other sources’ zero/negative-input extensions need separate definitions. Verify manual transcription against scans when necessary; Book Isle supports reading and retrieval, not execution of the algorithm.

## 2. Check both the division identity and remainder bounds

First divide forty-eight by eighteen: quotient two, remainder twelve. Check both reconstruction and zero≤remainder\<divisor. Quotient one with remainder thirty reconstructs forty-eight but violates the usual remainder bound. Record quotient, remainder and bounds separately rather than relying on an unexplained modulo symbol. Do not attempt division by zero; check stopping before the next division. Preserve the original calculation and correction reasons.

## 3. Map each next pair to divisor and remainder

The [NIST Euclidean algorithm definition](<https://xlinux.nist.gov/dads/HTML/euclidalgo.html>) computes the GCD of two positive integers by remainder iteration. The next pair after the first step is eighteen/twelve, not forty-eight/twelve; the following pair is twelve/six. Label all roles and the transfer to the next row. The multiplication symbol below is mathematical notation, not an app command.

```
48 = 2*18 + 12
18 = 1*12 + 6
12 = 2*6 + 0
Pairs: (48,18) -> (18,12) -> (12,6) -> (6,0)
GCD = 6
```

## 4. Zero stops the chain rather than replacing the result

Twelve divided by six leaves zero; the pair six/zero stops iteration and returns six. Do not divide by zero or return the final written zero. Explain the retained nonzero value rather than a vague “take the last number” rule. Starting a separate control at eighteen/six stops after one division and still gives six; fewer steps need not mean missing work. Other equal-input boundary workflows depend on their stated conventions. List length alone does not certify algorithm completeness.

## 5. Verify through common divisibility

Six divides forty-eight into eight and eighteen into three. Their positive common divisors are one, two, three and six, with six largest; three’s divisibility alone does not answer the greatest-divisor question. Remainder steps preserve common divisors because a divisor of both inputs divides their remainder, while reconstruction gives the reverse relation. Keep that justification separate from this example’s arithmetic. Candidate checking helps detect transcription errors without replacing the algorithm’s rationale.

## 6. Treat common multiples as a different question

Record LCM separately as 144, three times forty-eight and eight times eighteen. For positive integers it can be checked as the product divided by the GCD, but the remainder-chain answer remains six. GCD divides the inputs; LCM is divisible by them. Explain divisibility direction before using abbreviations rather than calling both a common number. This is a manual conceptual control, not another automatic app feature or a replacement for the original answer.

## 7. Troubleshoot missing transfers and invalid remainder bounds

For a nonterminating chain, check remainder bounds and pair transfer before calling it ordinary Euclidean iteration. For answer disagreement, inspect inputs, quotient and reconstruction before comparing an LCM answer. Resolve zero/O from the source instead of guessing a preferred stopping point. Open exported notes and verify divisions, pair arrows and returned value, not merely a visible six. Retrieve the passage by chapter/phrase rather than reimporting, keeping personal calculations separate from quotations.

## 8. Rebuild the chain and explain the returned object

Return to forty-eight/eighteen, cover results and rebuild remainders twelve, six, zero and each next pair. Explain returning six without another zero division. Recheck divisor/multiple direction and keep 144 out of the algorithm endpoint. Save edition/location, input domain, divisions, bounds, stopping state and independent checks. Completion is repeatable state transfer, not automatic proof of every integer case. Re-read changed input conventions and algorithm forms instead of inheriting old boundary assumptions.

## Related tutorials and sources

[Algorithm hand tracing](<https://allluckly.cn/en/guides/nasbook-appendix-algorithm-handtrace/>), [Counting rules](<https://allluckly.cn/en/guides/nasbook-combinations-role-permutations-four-people/>).

Checked on November 12, 2026 against the [official Book Isle App Store listing](<https://apps.apple.com/cn/app/id6766198898>) and [Apple public lookup](<https://itunes.apple.com/lookup?id=6766198898&country=cn&entity=software&timestamp=20261112>): Book Isle 1.1.4, requiring iOS / iPadOS 15.6 or later. Use existing reading, available notes or external records and verify controls in your installed version. Original samples and manual calculations are not new automatic analysis features. No physical-device, private-file or universal export-environment testing is claimed.