# Question 18890: Zero Inputs Have an Explicit Gcd Convention without Division by Zero

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0036-18890 · q-20261014-advance-20261105-r03-euclidean-algorithm-0090



## Original question and givens

Use original integers a=-15, b=0. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (a, b, gcd, x, y, linear_combination, convention).


- G001 — Use original integers a=-15, b=0. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (a, b, gcd, x, y, linear_combination, convention).





## Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.


- [mko-euclidean-algorithm](https://ocme.evemisslab.com/en/mko/mko-euclidean-algorithm/)





## Step-by-step derivation

S001 · impliesBefore

source operation=euclid-zero-input-boundary;original snapshot and mathematics retained

After

explicit ordered source output fields=[a, b, gcd, x, y, linear_combination, convention]

Conditions: none

Keep the historical interface marker in the original snapshot while listing the original ordered fields here.No JavaScript or external table is needed for the main text,and field labels are not mathematical evidence.

Step identifiers and dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001





S002 · impliesBefore

original signed inputs a=-15,b=0

After

initial nonnegative chain values=[15, 0];gcd(a,b)=gcd(abs(a),abs(b));never divide by a zero right input

Conditions: none

Divisibility is unaffected by input signs,and the gcd is nonnegative.Ordinary chain/invariant families require b≠0;the zero boundary separately retains its(0,0)convention.

Step identifiers and dependenciesRule: euclid_signed_inputs_absolute_chain_scope · Uses: G001, S001





S003 · impliesBefore

complete division rows=[];terminal pair=[15, 0]

After

gcd(a,b)=gcd(15,0)=15;all divisions used positive divisors;nonnegative gcd convention

Conditions: none

At termination common divisors are those of the terminal left value,giving its nonnegative gcd.The(0,0)value0 is an explicit separate convention.Do not append a fictitious division by zero.

Step identifiers and dependenciesRule: euclid_terminal_gcd_not_extra_zero_division · Uses: G001, S002





S004 · calculationBefore

Euclid restore original a sign in coefficient := mul(-1, 1)

After

Euclid restore original a sign in coefficient = -1

Conditions: none

Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003





S005 · calculationBefore

Euclid restore original b sign in coefficient := mul(1, 0)

After

Euclid restore original b sign in coefficient = 0

Conditions: none

Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004





S006 · calculationBefore

Euclid ORIGINAL signed a times x := mul(-15, -1)

After

Euclid ORIGINAL signed a times x = 15

Conditions: none

Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005





S007 · calculationBefore

Euclid ORIGINAL signed b times y := mul(0, 0)

After

Euclid ORIGINAL signed b times y = 0

Conditions: none

Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006





S008 · calculationBefore

Euclid final signed Bezout linear combination := sum(15, 0)

After

Euclid final signed Bezout linear combination = 15

Conditions: none

Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007





S009 · impliesBefore

original a=-15,b=0;deterministic absolute-chain coefficients=[1, 0]

After

sign-restored source coefficients x=-1,y=0;a*x+b*y=15=nonnegative gcd

Conditions: none

Retain the deterministic coefficients of this source extended chain even though other Bezout pairs may be valid.Use sign+1 for zero;(0,0)keeps(1,0)and0=0,without a uniqueness claim.

Step identifiers and dependenciesRule: euclid_deterministic_signed_bezout_not_just_any_coefficients · Uses: G001, S008









## Reverse checks


- S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S008 · premise_and_result_audit — Recompute Euclid final signed Bezout linear combination with exact rational arithmetic, not rounded decimals.
- S007 · premise_and_result_audit — Recompute Euclid ORIGINAL signed b times y with exact rational arithmetic, not rounded decimals.
- S006 · premise_and_result_audit — Recompute Euclid ORIGINAL signed a times x with exact rational arithmetic, not rounded decimals.
- S005 · premise_and_result_audit — Recompute Euclid restore original b sign in coefficient with exact rational arithmetic, not rounded decimals.
- S004 · premise_and_result_audit — Recompute Euclid restore original a sign in coefficient with exact rational arithmetic, not rounded decimals.
- S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.





## Result

(-15, 0, 15, -1, 0, 15, no_division_by_zero_executed)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.





## This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: 3ecca0c045eacb96e437d9a3daf9cb20ff3bdeefdfd576c534db4e1aefb00748
- candidate record SHA-256: 3d1a53e52a27f150f84b8604e4cd364558436bb422d181cf7582e1d6b6094d64
- semantic review receipt SHA-256: efee5462991619d6ed9f64f11ddb085e5614ef33f9f8fcd8cf29b9329f8fd301
- question-first witness SHA-256: acf91fa1622f01d2cf04b2c513294254323665351f660f2628aaafdb869c4ff2
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0090.json)