Question 18862: Gcd Preservation and Termination Have Different Obligations

Original question and givens

Use original integers a=1407, b=108. 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 (gcd_pairs, gcd, invariant, certificate).

• G001 — Use original integers a=1407, b=108. 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 (gcd_pairs, gcd, invariant, certificate).

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

Step-by-step derivation

source operation=euclid-gcd-invariant;original snapshot and mathematics retained

explicit ordered source output fields=[gcd_pairs, gcd, invariant, certificate]

original signed inputs a=1407,b=108

initial nonnegative chain values=[1407, 108];gcd(a,b)=gcd(abs(a),abs(b));never divide by a zero right input

Euclid step0 exact ratio := div(1407, 108)

Euclid step0 exact ratio = 469/36

Euclid step0 quotient times divisor := mul(13, 108)

Euclid step0 quotient times divisor = 1404

Euclid step0 remainder := sub(1407, 1404)

Euclid step0 remainder = 3

step=0;left=1407;positive right=108;exact ratio=469/36;floor quotient=13

left=right*q+r with r=3 in[0,108);gcd pair=[3, 3]

Euclid step1 exact ratio := div(108, 3)

Euclid step1 exact ratio = 36

Euclid step1 quotient times divisor := mul(36, 3)

Euclid step1 quotient times divisor = 108

Euclid step1 remainder := sub(108, 108)

Euclid step1 remainder = 0

step=1;left=108;positive right=3;exact ratio=36;floor quotient=36

left=right*q+r with r=0 in[0,3);gcd pair=[3, 3]

complete division rows=[[1407, 108, 13, 3], [108, 3, 36, 0]];terminal pair=[3, 0]

gcd(a,b)=gcd(3,0)=3;all divisions used positive divisors;nonnegative gcd convention

Reverse checks

• 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S009 · premise_and_result_audit — Recompute Euclid step1 remainder with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute Euclid step1 quotient times divisor with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute Euclid step1 exact ratio 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

([[3, 3], [3, 3]], 3, Yes, gcd_invariant_separate_from_termination)

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.