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Question 18895: Zero Inputs Have an Explicit Gcd Convention without Division by Zero

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0036-18895 · q-20261014-advance-20261105-r03-euclidean-algorithm-0095

Original question and givens

Use original integers a=0, b=20. 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=0, b=20. 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.

Step-by-step derivation

S001 · implies

Before

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 dependencies

Rule: source_ordered_interface_explicit_without_javascript · Uses: G001

S002 · implies

Before

original signed inputs a=0,b=20

After

initial nonnegative chain values=[0, 20];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 dependencies

Rule: euclid_signed_inputs_absolute_chain_scope · Uses: G001, S001

S003 · calculation

Before

Euclid step0 exact ratio := div(0, 20)

After

Euclid step0 exact ratio = 0

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · calculation

Before

Euclid step0 quotient times divisor := mul(0, 20)

After

Euclid step0 quotient times divisor = 0

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · calculation

Before

Euclid step0 remainder := sub(0, 0)

After

Euclid step0 remainder = 0

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · implies

Before

step=0;left=0;positive right=20;exact ratio=0;floor quotient=0

After

left=right*q+r with r=0 in[0,20);gcd pair=[20, 20]

Conditions: none

An integer d divides both left and right iff it divides right and left−q*right,so the gcd is unchanged.The next nonnegative right is the strictly smaller remainder,so infinite descent is impossible;termination is separate from invariance.

Step identifiers and dependencies

Rule: euclid_each_division_common_divisors_and_strict_descent · Uses: G001, S005

S007 · calculation

Before

Euclid step0 first coefficient multiple := mul(0, 0)

After

Euclid step0 first coefficient multiple = 0

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · calculation

Before

Euclid step0 first coefficient difference := sub(1, 0)

After

Euclid step0 first coefficient difference = 1

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S007

S009 · calculation

Before

Euclid step0 second coefficient multiple := mul(0, 1)

After

Euclid step0 second coefficient multiple = 0

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · calculation

Before

Euclid step0 second coefficient difference := sub(0, 0)

After

Euclid step0 second coefficient difference = 0

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S009

S011 · implies

Before

step=0;two original-absolute-input coefficient pairs=[[1, 0], [0, 1]];same quotient=0

After

next remainder coefficient pair=[1, 0];abs(a)*s_next+abs(b)*t_next=0

Conditions: none

Update both coefficient sequences with the same quotient as the remainder.Inductively every row remains a combination of the original absolute inputs;do not guess coefficients after finding the gcd.

Step identifiers and dependencies

Rule: euclid_extended_coefficients_same_quotient_recurrence · Uses: G001, S010

S012 · implies

Before

complete division rows=[[0, 20, 0, 0]];terminal pair=[20, 0]

After

gcd(a,b)=gcd(20,0)=20;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 dependencies

Rule: euclid_terminal_gcd_not_extra_zero_division · Uses: G001, S011

S013 · calculation

Before

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

After

Euclid restore original a sign in coefficient = 0

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S012

S014 · calculation

Before

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

After

Euclid restore original b sign in coefficient = 1

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S013

S015 · calculation

Before

Euclid ORIGINAL signed a times x := mul(0, 0)

After

Euclid ORIGINAL signed a times x = 0

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S014

S016 · calculation

Before

Euclid ORIGINAL signed b times y := mul(20, 1)

After

Euclid ORIGINAL signed b times y = 20

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S015

S017 · calculation

Before

Euclid final signed Bezout linear combination := sum(0, 20)

After

Euclid final signed Bezout linear combination = 20

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S016

S018 · implies

Before

original a=0,b=20;deterministic absolute-chain coefficients=[0, 1]

After

sign-restored source coefficients x=0,y=1;a*x+b*y=20=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 dependencies

Rule: euclid_deterministic_signed_bezout_not_just_any_coefficients · Uses: G001, S017

Reverse checks

  • S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S017 · premise_and_result_audit — Recompute Euclid final signed Bezout linear combination with exact rational arithmetic, not rounded decimals.
  • S016 · premise_and_result_audit — Recompute Euclid ORIGINAL signed b times y with exact rational arithmetic, not rounded decimals.
  • S015 · premise_and_result_audit — Recompute Euclid ORIGINAL signed a times x with exact rational arithmetic, not rounded decimals.
  • S014 · premise_and_result_audit — Recompute Euclid restore original b sign in coefficient with exact rational arithmetic, not rounded decimals.
  • S013 · premise_and_result_audit — Recompute Euclid restore original a sign in coefficient with exact rational arithmetic, not rounded decimals.
  • S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S010 · premise_and_result_audit — Recompute Euclid step0 second coefficient difference with exact rational arithmetic, not rounded decimals.
  • S009 · premise_and_result_audit — Recompute Euclid step0 second coefficient multiple with exact rational arithmetic, not rounded decimals.
  • S008 · premise_and_result_audit — Recompute Euclid step0 first coefficient difference with exact rational arithmetic, not rounded decimals.
  • S007 · premise_and_result_audit — Recompute Euclid step0 first coefficient multiple with exact rational arithmetic, not rounded decimals.
  • S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S005 · premise_and_result_audit — Recompute Euclid step0 remainder with exact rational arithmetic, not rounded decimals.
  • S004 · premise_and_result_audit — Recompute Euclid step0 quotient times divisor with exact rational arithmetic, not rounded decimals.
  • S003 · premise_and_result_audit — Recompute Euclid step0 exact ratio with exact rational arithmetic, not rounded decimals.
  • 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

(0, 20, 20, 0, 1, 20, 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: 8b1b96439e83748c1a6716bb43638f7951e910e8e270bb6a8722cdff1e278c2d
  • candidate record SHA-256: d36ac739f5fa82ac5489c2e67c7b67294093c4f3115714a615adf72439805119
  • semantic review receipt SHA-256: efee5462991619d6ed9f64f11ddb085e5614ef33f9f8fcd8cf29b9329f8fd301
  • question-first witness SHA-256: acf91fa1622f01d2cf04b2c513294254323665351f660f2628aaafdb869c4ff2
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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