Question 18875: Gcd Preservation and Termination Have Different Obligations

Original question and givens

Use original integers a=1888, b=121. 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=1888, b=121. 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=1888,b=121

initial nonnegative chain values=[1888, 121];gcd(a,b)=gcd(abs(a),abs(b));never divide by a zero right input

Euclid step0 exact ratio := div(1888, 121)

Euclid step0 exact ratio = 1888/121

Euclid step0 quotient times divisor := mul(15, 121)

Euclid step0 quotient times divisor = 1815

Euclid step0 remainder := sub(1888, 1815)

Euclid step0 remainder = 73

step=0;left=1888;positive right=121;exact ratio=1888/121;floor quotient=15

left=right*q+r with r=73 in[0,121);gcd pair=[1, 1]

Euclid step1 exact ratio := div(121, 73)

Euclid step1 exact ratio = 121/73

Euclid step1 quotient times divisor := mul(1, 73)

Euclid step1 quotient times divisor = 73

Euclid step1 remainder := sub(121, 73)

Euclid step1 remainder = 48

step=1;left=121;positive right=73;exact ratio=121/73;floor quotient=1

left=right*q+r with r=48 in[0,73);gcd pair=[1, 1]

Euclid step2 exact ratio := div(73, 48)

Euclid step2 exact ratio = 73/48

Euclid step2 quotient times divisor := mul(1, 48)

Euclid step2 quotient times divisor = 48

Euclid step2 remainder := sub(73, 48)

Euclid step2 remainder = 25

step=2;left=73;positive right=48;exact ratio=73/48;floor quotient=1

left=right*q+r with r=25 in[0,48);gcd pair=[1, 1]

Euclid step3 exact ratio := div(48, 25)

Euclid step3 exact ratio = 48/25

Euclid step3 quotient times divisor := mul(1, 25)

Euclid step3 quotient times divisor = 25

Euclid step3 remainder := sub(48, 25)

Euclid step3 remainder = 23

step=3;left=48;positive right=25;exact ratio=48/25;floor quotient=1

left=right*q+r with r=23 in[0,25);gcd pair=[1, 1]

Euclid step4 exact ratio := div(25, 23)

Euclid step4 exact ratio = 25/23

Euclid step4 quotient times divisor := mul(1, 23)

Euclid step4 quotient times divisor = 23

Euclid step4 remainder := sub(25, 23)

Euclid step4 remainder = 2

step=4;left=25;positive right=23;exact ratio=25/23;floor quotient=1

left=right*q+r with r=2 in[0,23);gcd pair=[1, 1]

Euclid step5 exact ratio := div(23, 2)

Euclid step5 exact ratio = 23/2

Euclid step5 quotient times divisor := mul(11, 2)

Euclid step5 quotient times divisor = 22

Euclid step5 remainder := sub(23, 22)

Euclid step5 remainder = 1

step=5;left=23;positive right=2;exact ratio=23/2;floor quotient=11

left=right*q+r with r=1 in[0,2);gcd pair=[1, 1]

Euclid step6 exact ratio := div(2, 1)

Euclid step6 exact ratio = 2

Euclid step6 quotient times divisor := mul(2, 1)

Euclid step6 quotient times divisor = 2

Euclid step6 remainder := sub(2, 2)

Euclid step6 remainder = 0

step=6;left=2;positive right=1;exact ratio=2;floor quotient=2

left=right*q+r with r=0 in[0,1);gcd pair=[1, 1]

complete division rows=[[1888, 121, 15, 73], [121, 73, 1, 48], [73, 48, 1, 25], [48, 25, 1, 23], [25, 23, 1, 2], [23, 2, 11, 1], [2, 1, 2, 0]];terminal pair=[1, 0]

gcd(a,b)=gcd(1,0)=1;all divisions used positive divisors;nonnegative gcd convention

Reverse checks

• S031 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S030 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S029 · premise_and_result_audit — Recompute Euclid step6 remainder with exact rational arithmetic, not rounded decimals.
• S028 · premise_and_result_audit — Recompute Euclid step6 quotient times divisor with exact rational arithmetic, not rounded decimals.
• S027 · premise_and_result_audit — Recompute Euclid step6 exact ratio with exact rational arithmetic, not rounded decimals.
• S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S025 · premise_and_result_audit — Recompute Euclid step5 remainder with exact rational arithmetic, not rounded decimals.
• S024 · premise_and_result_audit — Recompute Euclid step5 quotient times divisor with exact rational arithmetic, not rounded decimals.
• S023 · premise_and_result_audit — Recompute Euclid step5 exact ratio with exact rational arithmetic, not rounded decimals.
• S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S021 · premise_and_result_audit — Recompute Euclid step4 remainder with exact rational arithmetic, not rounded decimals.
• S020 · premise_and_result_audit — Recompute Euclid step4 quotient times divisor with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute Euclid step4 exact ratio with exact rational arithmetic, not rounded decimals.
• 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 step3 remainder with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute Euclid step3 quotient times divisor with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute Euclid step3 exact ratio with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S013 · premise_and_result_audit — Recompute Euclid step2 remainder with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute Euclid step2 quotient times divisor with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute Euclid step2 exact ratio with exact rational arithmetic, not rounded decimals.
• 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

([[1, 1], [1, 1], [1, 1], [1, 1], [1, 1], [1, 1], [1, 1]], 1, 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.