Original question and givens
Use original integers a=-21, b=0. 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=-21, b=0. 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).
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=-21,b=0
After
initial nonnegative chain values=[21, 0];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 · implies
Before
complete division rows=[];terminal pair=[21, 0]
After
gcd(a,b)=gcd(21,0)=21;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, S002
S004 · calculation
Before
Euclid restore original a sign in coefficient := mul(-1, 1)
After
Euclid restore original a 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, S003
S005 · calculation
Before
Euclid restore original b sign in coefficient := mul(1, 0)
After
Euclid restore original b 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, S004
S006 · calculation
Before
Euclid ORIGINAL signed a times x := mul(-21, -1)
After
Euclid ORIGINAL signed a times x = 21
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, S005
S007 · calculation
Before
Euclid ORIGINAL signed b times y := mul(0, 0)
After
Euclid ORIGINAL signed b times y = 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 final signed Bezout linear combination := sum(21, 0)
After
Euclid final signed Bezout linear combination = 21
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 · implies
Before
original a=-21,b=0;deterministic absolute-chain coefficients=[1, 0]
After
sign-restored source coefficients x=-1,y=0;a*x+b*y=21=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, S008
Result
(-21, 0, 21, -1, 0, 21, no_division_by_zero_executed)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.