Question 18826: Exact Quotient-Remainder Steps and Strict Descent Are Both Required

Original question and givens

Use original integers a=1000, b=97. 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 (division_chain, gcd, chain_valid, certificate).

• G001 — Use original integers a=1000, b=97. 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 (division_chain, gcd, chain_valid, 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-division-chain;original snapshot and mathematics retained

explicit ordered source output fields=[division_chain, gcd, chain_valid, certificate]

original signed inputs a=1000,b=97

initial nonnegative chain values=[1000, 97];gcd(a,b)=gcd(abs(a),abs(b));never divide by a zero right input

Euclid step0 exact ratio := div(1000, 97)

Euclid step0 exact ratio = 1000/97

Euclid step0 quotient times divisor := mul(10, 97)

Euclid step0 quotient times divisor = 970

Euclid step0 remainder := sub(1000, 970)

Euclid step0 remainder = 30

step=0;left=1000;positive right=97;exact ratio=1000/97;floor quotient=10

left=right*q+r with r=30 in[0,97);gcd pair=[1, 1]

Euclid step1 exact ratio := div(97, 30)

Euclid step1 exact ratio = 97/30

Euclid step1 quotient times divisor := mul(3, 30)

Euclid step1 quotient times divisor = 90

Euclid step1 remainder := sub(97, 90)

Euclid step1 remainder = 7

step=1;left=97;positive right=30;exact ratio=97/30;floor quotient=3

left=right*q+r with r=7 in[0,30);gcd pair=[1, 1]

Euclid step2 exact ratio := div(30, 7)

Euclid step2 exact ratio = 30/7

Euclid step2 quotient times divisor := mul(4, 7)

Euclid step2 quotient times divisor = 28

Euclid step2 remainder := sub(30, 28)

Euclid step2 remainder = 2

step=2;left=30;positive right=7;exact ratio=30/7;floor quotient=4

left=right*q+r with r=2 in[0,7);gcd pair=[1, 1]

Euclid step3 exact ratio := div(7, 2)

Euclid step3 exact ratio = 7/2

Euclid step3 quotient times divisor := mul(3, 2)

Euclid step3 quotient times divisor = 6

Euclid step3 remainder := sub(7, 6)

Euclid step3 remainder = 1

step=3;left=7;positive right=2;exact ratio=7/2;floor quotient=3

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

Euclid step4 exact ratio := div(2, 1)

Euclid step4 exact ratio = 2

Euclid step4 quotient times divisor := mul(2, 1)

Euclid step4 quotient times divisor = 2

Euclid step4 remainder := sub(2, 2)

Euclid step4 remainder = 0

step=4;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=[[1000, 97, 10, 30], [97, 30, 3, 7], [30, 7, 4, 2], [7, 2, 3, 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

• S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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

([[1000, 97, 10, 30], [97, 30, 3, 7], [30, 7, 4, 2], [7, 2, 3, 1], [2, 1, 2, 0]], 1, Yes, exact_quotient_remainder_and_strict_descent)

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.