# Question 18816: Restore Input Signs in the Extended Euclidean Witness

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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.

R0036-18816 · q-20261014-advance-20261105-r03-euclidean-algorithm-0066



## Original question and givens

Use original integers a=-746, b=119. 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, x, y, linear_combination, equal, certificate).


- G001 — Use original integers a=-746, b=119. 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, x, y, linear_combination, equal, 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](https://ocme.evemisslab.com/en/mko/mko-euclidean-algorithm/)





## Step-by-step derivation

S001 · impliesBefore

source operation=euclid-bezout-certificate;original snapshot and mathematics retained

After

explicit ordered source output fields=[gcd, x, y, linear_combination, equal, certificate]

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 dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001





S002 · impliesBefore

original signed inputs a=-746,b=119

After

initial nonnegative chain values=[746, 119];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 dependenciesRule: euclid_signed_inputs_absolute_chain_scope · Uses: G001, S001





S003 · calculationBefore

Euclid step0 exact ratio := div(746, 119)

After

Euclid step0 exact ratio = 746/119

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002





S004 · calculationBefore

Euclid step0 quotient times divisor := mul(6, 119)

After

Euclid step0 quotient times divisor = 714

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003





S005 · calculationBefore

Euclid step0 remainder := sub(746, 714)

After

Euclid step0 remainder = 32

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004





S006 · impliesBefore

step=0;left=746;positive right=119;exact ratio=746/119;floor quotient=6

After

left=right*q+r with r=32 in[0,119);gcd pair=[1, 1]

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 dependenciesRule: euclid_each_division_common_divisors_and_strict_descent · Uses: G001, S005





S007 · calculationBefore

Euclid step0 first coefficient multiple := mul(6, 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S006





S008 · calculationBefore

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S007





S009 · calculationBefore

Euclid step0 second coefficient multiple := mul(6, 1)

After

Euclid step0 second coefficient multiple = 6

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008





S010 · calculationBefore

Euclid step0 second coefficient difference := sub(0, 6)

After

Euclid step0 second coefficient difference = -6

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009





S011 · impliesBefore

step=0;two original-absolute-input coefficient pairs=[[1, 0], [0, 1]];same quotient=6

After

next remainder coefficient pair=[1, -6];abs(a)*s_next+abs(b)*t_next=32

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 dependenciesRule: euclid_extended_coefficients_same_quotient_recurrence · Uses: G001, S010





S012 · calculationBefore

Euclid step1 exact ratio := div(119, 32)

After

Euclid step1 exact ratio = 119/32

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011





S013 · calculationBefore

Euclid step1 quotient times divisor := mul(3, 32)

After

Euclid step1 quotient times divisor = 96

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012





S014 · calculationBefore

Euclid step1 remainder := sub(119, 96)

After

Euclid step1 remainder = 23

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013





S015 · impliesBefore

step=1;left=119;positive right=32;exact ratio=119/32;floor quotient=3

After

left=right*q+r with r=23 in[0,32);gcd pair=[1, 1]

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 dependenciesRule: euclid_each_division_common_divisors_and_strict_descent · Uses: G001, S014





S016 · calculationBefore

Euclid step1 first coefficient multiple := mul(3, 1)

After

Euclid step1 first coefficient multiple = 3

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015





S017 · calculationBefore

Euclid step1 first coefficient difference := sub(0, 3)

After

Euclid step1 first coefficient difference = -3

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016





S018 · calculationBefore

Euclid step1 second coefficient multiple := mul(3, -6)

After

Euclid step1 second coefficient multiple = -18

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017





S019 · calculationBefore

Euclid step1 second coefficient difference := sub(1, -18)

After

Euclid step1 second coefficient difference = 19

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018





S020 · impliesBefore

step=1;two original-absolute-input coefficient pairs=[[0, 1], [1, -6]];same quotient=3

After

next remainder coefficient pair=[-3, 19];abs(a)*s_next+abs(b)*t_next=23

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 dependenciesRule: euclid_extended_coefficients_same_quotient_recurrence · Uses: G001, S019





S021 · calculationBefore

Euclid step2 exact ratio := div(32, 23)

After

Euclid step2 exact ratio = 32/23

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020





S022 · calculationBefore

Euclid step2 quotient times divisor := mul(1, 23)

After

Euclid step2 quotient times divisor = 23

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021





S023 · calculationBefore

Euclid step2 remainder := sub(32, 23)

After

Euclid step2 remainder = 9

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S022





S024 · impliesBefore

step=2;left=32;positive right=23;exact ratio=32/23;floor quotient=1

After

left=right*q+r with r=9 in[0,23);gcd pair=[1, 1]

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 dependenciesRule: euclid_each_division_common_divisors_and_strict_descent · Uses: G001, S023





S025 · calculationBefore

Euclid step2 first coefficient multiple := mul(1, -3)

After

Euclid step2 first coefficient multiple = -3

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S024





S026 · calculationBefore

Euclid step2 first coefficient difference := sub(1, -3)

After

Euclid step2 first coefficient difference = 4

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S025





S027 · calculationBefore

Euclid step2 second coefficient multiple := mul(1, 19)

After

Euclid step2 second coefficient multiple = 19

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S026





S028 · calculationBefore

Euclid step2 second coefficient difference := sub(-6, 19)

After

Euclid step2 second coefficient difference = -25

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S027





S029 · impliesBefore

step=2;two original-absolute-input coefficient pairs=[[1, -6], [-3, 19]];same quotient=1

After

next remainder coefficient pair=[4, -25];abs(a)*s_next+abs(b)*t_next=9

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 dependenciesRule: euclid_extended_coefficients_same_quotient_recurrence · Uses: G001, S028





S030 · calculationBefore

Euclid step3 exact ratio := div(23, 9)

After

Euclid step3 exact ratio = 23/9

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S029





S031 · calculationBefore

Euclid step3 quotient times divisor := mul(2, 9)

After

Euclid step3 quotient times divisor = 18

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S030





S032 · calculationBefore

Euclid step3 remainder := sub(23, 18)

After

Euclid step3 remainder = 5

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S031





S033 · impliesBefore

step=3;left=23;positive right=9;exact ratio=23/9;floor quotient=2

After

left=right*q+r with r=5 in[0,9);gcd pair=[1, 1]

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 dependenciesRule: euclid_each_division_common_divisors_and_strict_descent · Uses: G001, S032





S034 · calculationBefore

Euclid step3 first coefficient multiple := mul(2, 4)

After

Euclid step3 first coefficient multiple = 8

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S033





S035 · calculationBefore

Euclid step3 first coefficient difference := sub(-3, 8)

After

Euclid step3 first coefficient difference = -11

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S034





S036 · calculationBefore

Euclid step3 second coefficient multiple := mul(2, -25)

After

Euclid step3 second coefficient multiple = -50

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S035





S037 · calculationBefore

Euclid step3 second coefficient difference := sub(19, -50)

After

Euclid step3 second coefficient difference = 69

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S036





S038 · impliesBefore

step=3;two original-absolute-input coefficient pairs=[[-3, 19], [4, -25]];same quotient=2

After

next remainder coefficient pair=[-11, 69];abs(a)*s_next+abs(b)*t_next=5

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 dependenciesRule: euclid_extended_coefficients_same_quotient_recurrence · Uses: G001, S037





S039 · calculationBefore

Euclid step4 exact ratio := div(9, 5)

After

Euclid step4 exact ratio = 9/5

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S038





S040 · calculationBefore

Euclid step4 quotient times divisor := mul(1, 5)

After

Euclid step4 quotient times divisor = 5

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S039





S041 · calculationBefore

Euclid step4 remainder := sub(9, 5)

After

Euclid step4 remainder = 4

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S040





S042 · impliesBefore

step=4;left=9;positive right=5;exact ratio=9/5;floor quotient=1

After

left=right*q+r with r=4 in[0,5);gcd pair=[1, 1]

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 dependenciesRule: euclid_each_division_common_divisors_and_strict_descent · Uses: G001, S041





S043 · calculationBefore

Euclid step4 first coefficient multiple := mul(1, -11)

After

Euclid step4 first coefficient multiple = -11

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S042





S044 · calculationBefore

Euclid step4 first coefficient difference := sub(4, -11)

After

Euclid step4 first coefficient difference = 15

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S043





S045 · calculationBefore

Euclid step4 second coefficient multiple := mul(1, 69)

After

Euclid step4 second coefficient multiple = 69

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S044





S046 · calculationBefore

Euclid step4 second coefficient difference := sub(-25, 69)

After

Euclid step4 second coefficient difference = -94

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S045





S047 · impliesBefore

step=4;two original-absolute-input coefficient pairs=[[4, -25], [-11, 69]];same quotient=1

After

next remainder coefficient pair=[15, -94];abs(a)*s_next+abs(b)*t_next=4

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 dependenciesRule: euclid_extended_coefficients_same_quotient_recurrence · Uses: G001, S046





S048 · calculationBefore

Euclid step5 exact ratio := div(5, 4)

After

Euclid step5 exact ratio = 5/4

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S047





S049 · calculationBefore

Euclid step5 quotient times divisor := mul(1, 4)

After

Euclid step5 quotient times divisor = 4

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S048





S050 · calculationBefore

Euclid step5 remainder := sub(5, 4)

After

Euclid step5 remainder = 1

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S049





S051 · impliesBefore

step=5;left=5;positive right=4;exact ratio=5/4;floor quotient=1

After

left=right*q+r with r=1 in[0,4);gcd pair=[1, 1]

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 dependenciesRule: euclid_each_division_common_divisors_and_strict_descent · Uses: G001, S050





S052 · calculationBefore

Euclid step5 first coefficient multiple := mul(1, 15)

After

Euclid step5 first coefficient multiple = 15

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S051





S053 · calculationBefore

Euclid step5 first coefficient difference := sub(-11, 15)

After

Euclid step5 first coefficient difference = -26

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S052





S054 · calculationBefore

Euclid step5 second coefficient multiple := mul(1, -94)

After

Euclid step5 second coefficient multiple = -94

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S053





S055 · calculationBefore

Euclid step5 second coefficient difference := sub(69, -94)

After

Euclid step5 second coefficient difference = 163

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S054





S056 · impliesBefore

step=5;two original-absolute-input coefficient pairs=[[-11, 69], [15, -94]];same quotient=1

After

next remainder coefficient pair=[-26, 163];abs(a)*s_next+abs(b)*t_next=1

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 dependenciesRule: euclid_extended_coefficients_same_quotient_recurrence · Uses: G001, S055





S057 · calculationBefore

Euclid step6 exact ratio := div(4, 1)

After

Euclid step6 exact ratio = 4

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S056





S058 · calculationBefore

Euclid step6 quotient times divisor := mul(4, 1)

After

Euclid step6 quotient times divisor = 4

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S057





S059 · calculationBefore

Euclid step6 remainder := sub(4, 4)

After

Euclid step6 remainder = 0

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S058





S060 · impliesBefore

step=6;left=4;positive right=1;exact ratio=4;floor quotient=4

After

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

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 dependenciesRule: euclid_each_division_common_divisors_and_strict_descent · Uses: G001, S059





S061 · calculationBefore

Euclid step6 first coefficient multiple := mul(4, -26)

After

Euclid step6 first coefficient multiple = -104

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S060





S062 · calculationBefore

Euclid step6 first coefficient difference := sub(15, -104)

After

Euclid step6 first coefficient difference = 119

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S061





S063 · calculationBefore

Euclid step6 second coefficient multiple := mul(4, 163)

After

Euclid step6 second coefficient multiple = 652

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S062





S064 · calculationBefore

Euclid step6 second coefficient difference := sub(-94, 652)

After

Euclid step6 second coefficient difference = -746

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S063





S065 · impliesBefore

step=6;two original-absolute-input coefficient pairs=[[15, -94], [-26, 163]];same quotient=4

After

next remainder coefficient pair=[119, -746];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 dependenciesRule: euclid_extended_coefficients_same_quotient_recurrence · Uses: G001, S064





S066 · impliesBefore

complete division rows=[[746, 119, 6, 32], [119, 32, 3, 23], [32, 23, 1, 9], [23, 9, 2, 5], [9, 5, 1, 4], [5, 4, 1, 1], [4, 1, 4, 0]];terminal pair=[1, 0]

After

gcd(a,b)=gcd(1,0)=1;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 dependenciesRule: euclid_terminal_gcd_not_extra_zero_division · Uses: G001, S065





S067 · calculationBefore

Euclid restore original a sign in coefficient := mul(-1, -26)

After

Euclid restore original a sign in coefficient = 26

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S066





S068 · calculationBefore

Euclid restore original b sign in coefficient := mul(1, 163)

After

Euclid restore original b sign in coefficient = 163

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S067





S069 · calculationBefore

Euclid ORIGINAL signed a times x := mul(-746, 26)

After

Euclid ORIGINAL signed a times x = -19396

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S068





S070 · calculationBefore

Euclid ORIGINAL signed b times y := mul(119, 163)

After

Euclid ORIGINAL signed b times y = 19397

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S069





S071 · calculationBefore

Euclid final signed Bezout linear combination := sum(-19396, 19397)

After

Euclid final signed Bezout linear combination = 1

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S070





S072 · impliesBefore

original a=-746,b=119;deterministic absolute-chain coefficients=[-26, 163]

After

sign-restored source coefficients x=26,y=163;a*x+b*y=1=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 dependenciesRule: euclid_deterministic_signed_bezout_not_just_any_coefficients · Uses: G001, S071









## Reverse checks


- S072 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S071 · premise_and_result_audit — Recompute Euclid final signed Bezout linear combination with exact rational arithmetic, not rounded decimals.
- S070 · premise_and_result_audit — Recompute Euclid ORIGINAL signed b times y with exact rational arithmetic, not rounded decimals.
- S069 · premise_and_result_audit — Recompute Euclid ORIGINAL signed a times x with exact rational arithmetic, not rounded decimals.
- S068 · premise_and_result_audit — Recompute Euclid restore original b sign in coefficient with exact rational arithmetic, not rounded decimals.
- S067 · premise_and_result_audit — Recompute Euclid restore original a sign in coefficient with exact rational arithmetic, not rounded decimals.
- S066 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S065 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S064 · premise_and_result_audit — Recompute Euclid step6 second coefficient difference with exact rational arithmetic, not rounded decimals.
- S063 · premise_and_result_audit — Recompute Euclid step6 second coefficient multiple with exact rational arithmetic, not rounded decimals.
- S062 · premise_and_result_audit — Recompute Euclid step6 first coefficient difference with exact rational arithmetic, not rounded decimals.
- S061 · premise_and_result_audit — Recompute Euclid step6 first coefficient multiple with exact rational arithmetic, not rounded decimals.
- S060 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S059 · premise_and_result_audit — Recompute Euclid step6 remainder with exact rational arithmetic, not rounded decimals.
- S058 · premise_and_result_audit — Recompute Euclid step6 quotient times divisor with exact rational arithmetic, not rounded decimals.
- S057 · premise_and_result_audit — Recompute Euclid step6 exact ratio with exact rational arithmetic, not rounded decimals.
- S056 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S055 · premise_and_result_audit — Recompute Euclid step5 second coefficient difference with exact rational arithmetic, not rounded decimals.
- S054 · premise_and_result_audit — Recompute Euclid step5 second coefficient multiple with exact rational arithmetic, not rounded decimals.
- S053 · premise_and_result_audit — Recompute Euclid step5 first coefficient difference with exact rational arithmetic, not rounded decimals.
- S052 · premise_and_result_audit — Recompute Euclid step5 first coefficient multiple with exact rational arithmetic, not rounded decimals.
- S051 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S050 · premise_and_result_audit — Recompute Euclid step5 remainder with exact rational arithmetic, not rounded decimals.
- S049 · premise_and_result_audit — Recompute Euclid step5 quotient times divisor with exact rational arithmetic, not rounded decimals.
- S048 · premise_and_result_audit — Recompute Euclid step5 exact ratio with exact rational arithmetic, not rounded decimals.
- S047 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S046 · premise_and_result_audit — Recompute Euclid step4 second coefficient difference with exact rational arithmetic, not rounded decimals.
- S045 · premise_and_result_audit — Recompute Euclid step4 second coefficient multiple with exact rational arithmetic, not rounded decimals.
- S044 · premise_and_result_audit — Recompute Euclid step4 first coefficient difference with exact rational arithmetic, not rounded decimals.
- S043 · premise_and_result_audit — Recompute Euclid step4 first coefficient multiple with exact rational arithmetic, not rounded decimals.
- S042 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S041 · premise_and_result_audit — Recompute Euclid step4 remainder with exact rational arithmetic, not rounded decimals.
- S040 · premise_and_result_audit — Recompute Euclid step4 quotient times divisor with exact rational arithmetic, not rounded decimals.
- S039 · premise_and_result_audit — Recompute Euclid step4 exact ratio with exact rational arithmetic, not rounded decimals.
- S038 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S037 · premise_and_result_audit — Recompute Euclid step3 second coefficient difference with exact rational arithmetic, not rounded decimals.
- S036 · premise_and_result_audit — Recompute Euclid step3 second coefficient multiple with exact rational arithmetic, not rounded decimals.
- S035 · premise_and_result_audit — Recompute Euclid step3 first coefficient difference with exact rational arithmetic, not rounded decimals.
- S034 · premise_and_result_audit — Recompute Euclid step3 first coefficient multiple with exact rational arithmetic, not rounded decimals.
- S033 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S032 · premise_and_result_audit — Recompute Euclid step3 remainder with exact rational arithmetic, not rounded decimals.
- S031 · premise_and_result_audit — Recompute Euclid step3 quotient times divisor with exact rational arithmetic, not rounded decimals.
- S030 · premise_and_result_audit — Recompute Euclid step3 exact ratio with exact rational arithmetic, not rounded decimals.
- S029 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S028 · premise_and_result_audit — Recompute Euclid step2 second coefficient difference with exact rational arithmetic, not rounded decimals.
- S027 · premise_and_result_audit — Recompute Euclid step2 second coefficient multiple with exact rational arithmetic, not rounded decimals.
- S026 · premise_and_result_audit — Recompute Euclid step2 first coefficient difference with exact rational arithmetic, not rounded decimals.
- S025 · premise_and_result_audit — Recompute Euclid step2 first coefficient multiple with exact rational arithmetic, not rounded decimals.
- S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S023 · premise_and_result_audit — Recompute Euclid step2 remainder with exact rational arithmetic, not rounded decimals.
- S022 · premise_and_result_audit — Recompute Euclid step2 quotient times divisor with exact rational arithmetic, not rounded decimals.
- S021 · premise_and_result_audit — Recompute Euclid step2 exact ratio with exact rational arithmetic, not rounded decimals.
- S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S019 · premise_and_result_audit — Recompute Euclid step1 second coefficient difference with exact rational arithmetic, not rounded decimals.
- S018 · premise_and_result_audit — Recompute Euclid step1 second coefficient multiple with exact rational arithmetic, not rounded decimals.
- S017 · premise_and_result_audit — Recompute Euclid step1 first coefficient difference with exact rational arithmetic, not rounded decimals.
- S016 · premise_and_result_audit — Recompute Euclid step1 first coefficient multiple with exact rational arithmetic, not rounded decimals.
- S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S014 · premise_and_result_audit — Recompute Euclid step1 remainder with exact rational arithmetic, not rounded decimals.
- S013 · premise_and_result_audit — Recompute Euclid step1 quotient times divisor with exact rational arithmetic, not rounded decimals.
- S012 · premise_and_result_audit — Recompute Euclid step1 exact ratio with exact rational arithmetic, not rounded decimals.
- 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

(1, 26, 163, 1, Yes, extended_euclidean_back_substitution)

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: 19f53f22f10786549d401d1bbeadecc6330f27a9c41366cb9a3662e5867c69a8
- candidate record SHA-256: f131d99f68892a92c50277957fcee892f798ab1a18717329705685e351ba5c41
- semantic review receipt SHA-256: efee5462991619d6ed9f64f11ddb085e5614ef33f9f8fcd8cf29b9329f8fd301
- question-first witness SHA-256: acf91fa1622f01d2cf04b2c513294254323665351f660f2628aaafdb869c4ff2
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261105-r03-euclidean-algorithm-0066.json)