# Question 19623: Double-Zero Coefficients Must Not Divide by Gcd Zero

[Return to catalogue page 99](https://ocme.evemisslab.com/en/questions/page/99/)

[← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations/#q-20261014-advance-20261106-r06-linear-diophantine-equation-0098)

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.

R0038-19623 · q-20261014-advance-20261106-r06-linear-diophantine-equation-0098



## Original question and givens

Use exact original integers a=0, b=0, c=10 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (c, solvable, solution_set, boundary).


- G001 — Use exact original integers a=0, b=0, c=10 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (c, solvable, solution_set, boundary).





## 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-linear-diophantine-equation](https://ocme.evemisslab.com/en/mko/mko-linear-diophantine-equation/)





## Step-by-step derivation

S001 · impliesBefore

source operation=dioph-degenerate-boundary;original snapshot and mathematics retained

After

explicit ordered source output fields=[c, solvable, solution_set, boundary]

Conditions: none

Keep the historical interface marker in the original snapshot while listing ordered source fields here,without JavaScript or external lookup.Field labels are not mathematical evidence,and No retains the operation's claim scope.

Step identifiers and dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001





S002 · impliesBefore

ACTUAL original equation0*x+0*y=10;BOTH coefficients0

After

left side is0 for EVERY integer pair;solvable iff c=0=No;solution set=no_integer_solution;NO gcd0 division

Conditions: none

With both coefficients zero the left side is identically0.For c=0 all of Z² solves it,not one parameter line;otherwise none does.Never divide by or reduce modulo gcd0.

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









## Reverse checks


- 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

(10, No, no_integer_solution, never_divide_by_gcd_zero)

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: 680095d89cc63c8e89639b7e110f93fb6618333b3584e85a51a7c01276a89e3e
- candidate record SHA-256: 1e1caabee109a76bb47eb05dee51989678ae9873a0e48652a17e60af1f50dda0
- semantic review receipt SHA-256: d165cad30849a3aba67390a59c572c0d965269d4b405e43df36da19e28d85612
- question-first witness SHA-256: c222ca47a07939c019865a1e4d9ce633205cdee68db92bf96a96f5683a62ebc7
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0098.json)