# Question 19660: Scale a Signed Bezout Pair to Obtain One Integer Solution [Return to catalogue page 99](https://ocme.evemisslab.com/en/questions/page/99/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261106-r06-linear-diophantine-equation-foundations/#q-20261014-advance-20261106-r06-linear-diophantine-equation-0035) 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. R0038-19660 · q-20261014-advance-20261106-r06-linear-diophantine-equation-0035 ## Original question and givens Use exact original integers a=-66, b=99, c=330 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (gcd, bezout_x, bezout_y, x0, y0, linear_combination, certificate_valid, certificate). - G001 — Use exact original integers a=-66, b=99, c=330 in a*x+b*y=c. The gcd d is nonnegative, and signed Bezout coefficients follow the deterministic extended Euclidean chain on absolute inputs, restoring input signs. For (a,b)≠(0,0), solvability is exactly d dividing c; finite searches do not supply all integer solutions. Handle (a,b)=(0,0) separately without division by d=0. Give (gcd, bezout_x, bezout_y, x0, y0, linear_combination, certificate_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-linear-diophantine-equation](https://ocme.evemisslab.com/en/mko/mko-linear-diophantine-equation/) ## Step-by-step derivation S001 · impliesBefore source operation=dioph-particular-solution;original snapshot and mathematics retained After explicit ordered source output fields=[gcd, bezout_x, bezout_y, x0, y0, linear_combination, certificate_valid, certificate] Conditions: none Keep the historical interface marker in the original snapshot while listing ordered source fields here,without JavaScript or external lookup.Field labels are not mathematical evidence,and No retains the operation's claim scope. Step identifiers and dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001 S002 · impliesBefore ORIGINAL signed coefficients a=-66,b=99 After start exact Euclidean chain on absolute values=[66, 99];nonnegative gcd;source deterministic coefficient recurrence Conditions: none Use absolute inputs for the nonnegative gcd,but restore original signs in Bezout coefficients.Retain this source deterministic extended-chain pair rather than substituting another valid but different pair. Step identifiers and dependenciesRule: dioph_original_signs_deterministic_extended_chain · Uses: G001, S001 S003 · calculationBefore Dioph Euclid step0 exact ratio := div(66, 99) After Dioph Euclid step0 exact ratio = 2/3 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore Dioph Euclid step0 divisor multiple := mul(0, 99) After Dioph Euclid step0 divisor multiple = 0 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore Dioph Euclid step0 remainder := sub(66, 0) After Dioph Euclid step0 remainder = 66 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore Dioph Euclid step0 first coefficient multiple := mul(0, 0) After Dioph Euclid step0 first coefficient multiple = 0 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore Dioph Euclid step0 first coefficient difference := sub(1, 0) After Dioph Euclid step0 first coefficient difference = 1 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore Dioph Euclid step0 second coefficient multiple := mul(0, 1) After Dioph Euclid step0 second coefficient multiple = 0 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore Dioph Euclid step0 second coefficient difference := sub(0, 0) After Dioph Euclid step0 second coefficient difference = 0 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · impliesBefore step=0;positive divisor=99;old pair=[66, 99];exact ratio=2/3;floor quotient=0;old coefficient pairs=[[1, 0], [0, 1]] After next remainder=66 in[0,99);next signed absolute-input coefficients=[1, 0];common divisor set unchanged Conditions: none Remainder subtraction preserves the common-divisor set,and the same quotient updates both coefficients so abs(a)*s+abs(b)*t equals the remainder.Nonnegative strict remainder descent guarantees termination without division by0. Step identifiers and dependenciesRule: dioph_every_extended_step_invariant_and_descent · Uses: G001, S009 S011 · calculationBefore Dioph Euclid step1 exact ratio := div(99, 66) After Dioph Euclid step1 exact ratio = 3/2 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore Dioph Euclid step1 divisor multiple := mul(1, 66) After Dioph Euclid step1 divisor multiple = 66 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore Dioph Euclid step1 remainder := sub(99, 66) After Dioph Euclid step1 remainder = 33 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore Dioph Euclid step1 first coefficient multiple := mul(1, 1) After Dioph Euclid step1 first coefficient multiple = 1 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore Dioph Euclid step1 first coefficient difference := sub(0, 1) After Dioph Euclid step1 first coefficient difference = -1 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · calculationBefore Dioph Euclid step1 second coefficient multiple := mul(1, 0) After Dioph Euclid step1 second coefficient multiple = 0 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · calculationBefore Dioph Euclid step1 second coefficient difference := sub(1, 0) After Dioph Euclid step1 second coefficient difference = 1 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · impliesBefore step=1;positive divisor=66;old pair=[99, 66];exact ratio=3/2;floor quotient=1;old coefficient pairs=[[0, 1], [1, 0]] After next remainder=33 in[0,66);next signed absolute-input coefficients=[-1, 1];common divisor set unchanged Conditions: none Remainder subtraction preserves the common-divisor set,and the same quotient updates both coefficients so abs(a)*s+abs(b)*t equals the remainder.Nonnegative strict remainder descent guarantees termination without division by0. Step identifiers and dependenciesRule: dioph_every_extended_step_invariant_and_descent · Uses: G001, S017 S019 · calculationBefore Dioph Euclid step2 exact ratio := div(66, 33) After Dioph Euclid step2 exact ratio = 2 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018 S020 · calculationBefore Dioph Euclid step2 divisor multiple := mul(2, 33) After Dioph Euclid step2 divisor multiple = 66 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · calculationBefore Dioph Euclid step2 remainder := sub(66, 66) After Dioph Euclid step2 remainder = 0 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020 S022 · calculationBefore Dioph Euclid step2 first coefficient multiple := mul(2, -1) After Dioph Euclid step2 first coefficient multiple = -2 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021 S023 · calculationBefore Dioph Euclid step2 first coefficient difference := sub(1, -2) After Dioph Euclid step2 first coefficient difference = 3 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S022 S024 · calculationBefore Dioph Euclid step2 second coefficient multiple := mul(2, 1) After Dioph Euclid step2 second coefficient multiple = 2 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S023 S025 · calculationBefore Dioph Euclid step2 second coefficient difference := sub(0, 2) After Dioph Euclid step2 second coefficient difference = -2 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S024 S026 · impliesBefore step=2;positive divisor=33;old pair=[66, 33];exact ratio=2;floor quotient=2;old coefficient pairs=[[1, 0], [-1, 1]] After next remainder=0 in[0,33);next signed absolute-input coefficients=[3, -2];common divisor set unchanged Conditions: none Remainder subtraction preserves the common-divisor set,and the same quotient updates both coefficients so abs(a)*s+abs(b)*t equals the remainder.Nonnegative strict remainder descent guarantees termination without division by0. Step identifiers and dependenciesRule: dioph_every_extended_step_invariant_and_descent · Uses: G001, S025 S027 · calculationBefore Dioph restore original a sign := mul(-1, -1) After Dioph restore original a sign = 1 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S026 S028 · calculationBefore Dioph restore original b sign := mul(1, 1) After Dioph restore original b sign = 1 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S027 S029 · calculationBefore Dioph ORIGINAL a times Bezout x := mul(-66, 1) After Dioph ORIGINAL a times Bezout x = -66 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S028 S030 · calculationBefore Dioph ORIGINAL b times Bezout y := mul(99, 1) After Dioph ORIGINAL b times Bezout y = 99 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S029 S031 · calculationBefore Dioph original signed Bezout identity := sum(-66, 99) After Dioph original signed Bezout identity = 33 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S030 S032 · impliesBefore complete exact chain=[[66, 99, 0, 66], [99, 66, 1, 33], [66, 33, 2, 0]];terminal pair=[33, 0];sign-restored x=1,y=1 After a*x+b*y=33=nonnegative gcd d=33;gcd from invariance and terminal gcd(d,0)=d Conditions: none The last nonzero remainder is the gcd;substitute the original signs to verify a*x+b*y=d.One zero coefficient is legitimate and creates no extra division-by-zero row. Step identifiers and dependenciesRule: dioph_terminal_gcd_and_original_signed_identity · Uses: G001, S031 S033 · calculationBefore Dioph target divided by positive gcd := div(330, 33) After Dioph target divided by positive gcd = 10 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S032 S034 · impliesBefore ACTUAL original equation-66*x+99*y=330;positive gcd=33;exact c/d=10;Bezout coefficients=[1, 1] After d divides both coefficients,so ANY integer solution requires d divides c;conversely integer c/d scales Bezout to a solution;solvable=Yes Conditions: none The gcd divides every integer linear combination,giving necessity.An integer c/d scales the displayed Bezout identity,giving sufficiency.A noninteger quotient obstructs all integer solutions,not merely a finite search range. Step identifiers and dependenciesRule: dioph_gcd_divisibility_iff_not_finite_search · Uses: G001, S033 S035 · calculationBefore Dioph particular x0 from scaled source Bezout := mul(1, 10) After Dioph particular x0 from scaled source Bezout = 10 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S034 S036 · calculationBefore Dioph particular y0 from scaled source Bezout := mul(1, 10) After Dioph particular y0 from scaled source Bezout = 10 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S035 S037 · calculationBefore Dioph original a times particular x0 := mul(-66, 10) After Dioph original a times particular x0 = -660 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S036 S038 · calculationBefore Dioph original b times particular y0 := mul(99, 10) After Dioph original b times particular y0 = 990 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S037 S039 · calculationBefore Dioph particular original equation check := sum(-660, 990) After Dioph particular original equation check = 330 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S038 ## Reverse checks - S039 · premise_and_result_audit — Recompute Dioph particular original equation check with exact rational arithmetic, not rounded decimals. - S038 · premise_and_result_audit — Recompute Dioph original b times particular y0 with exact rational arithmetic, not rounded decimals. - S037 · premise_and_result_audit — Recompute Dioph original a times particular x0 with exact rational arithmetic, not rounded decimals. - S036 · premise_and_result_audit — Recompute Dioph particular y0 from scaled source Bezout with exact rational arithmetic, not rounded decimals. - S035 · premise_and_result_audit — Recompute Dioph particular x0 from scaled source Bezout with exact rational arithmetic, not rounded decimals. - S034 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S033 · premise_and_result_audit — Recompute Dioph target divided by positive gcd with exact rational arithmetic, not rounded decimals. - S032 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S031 · premise_and_result_audit — Recompute Dioph original signed Bezout identity with exact rational arithmetic, not rounded decimals. - S030 · premise_and_result_audit — Recompute Dioph ORIGINAL b times Bezout y with exact rational arithmetic, not rounded decimals. - S029 · premise_and_result_audit — Recompute Dioph ORIGINAL a times Bezout x with exact rational arithmetic, not rounded decimals. - S028 · premise_and_result_audit — Recompute Dioph restore original b sign with exact rational arithmetic, not rounded decimals. - S027 · premise_and_result_audit — Recompute Dioph restore original a sign 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 Dioph Euclid step2 second coefficient difference with exact rational arithmetic, not rounded decimals. - S024 · premise_and_result_audit — Recompute Dioph Euclid step2 second coefficient multiple with exact rational arithmetic, not rounded decimals. - S023 · premise_and_result_audit — Recompute Dioph Euclid step2 first coefficient difference with exact rational arithmetic, not rounded decimals. - S022 · premise_and_result_audit — Recompute Dioph Euclid step2 first coefficient multiple with exact rational arithmetic, not rounded decimals. - S021 · premise_and_result_audit — Recompute Dioph Euclid step2 remainder with exact rational arithmetic, not rounded decimals. - S020 · premise_and_result_audit — Recompute Dioph Euclid step2 divisor multiple with exact rational arithmetic, not rounded decimals. - S019 · premise_and_result_audit — Recompute Dioph Euclid step2 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 Dioph Euclid step1 second coefficient difference with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Recompute Dioph Euclid step1 second coefficient multiple with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute Dioph Euclid step1 first coefficient difference with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute Dioph Euclid step1 first coefficient multiple with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute Dioph Euclid step1 remainder with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute Dioph Euclid step1 divisor multiple with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute Dioph Euclid step1 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 Dioph Euclid step0 second coefficient difference with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute Dioph Euclid step0 second coefficient multiple with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute Dioph Euclid step0 first coefficient difference with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute Dioph Euclid step0 first coefficient multiple with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute Dioph Euclid step0 remainder with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute Dioph Euclid step0 divisor multiple with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute Dioph 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 (33, 1, 1, 10, 10, 330, Yes, extended_Euclidean_Bezout_certificate) 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: 6d2dc24fe533dc34aa965be39823e6963dcf6e07afda98061190d42cab53075e - candidate record SHA-256: 9e7389248fe90894def019cdf606f006afbc750f7030335b2e0bb812445eb432 - semantic review receipt SHA-256: d165cad30849a3aba67390a59c572c0d965269d4b405e43df36da19e28d85612 - question-first witness SHA-256: c222ca47a07939c019865a1e4d9ce633205cdee68db92bf96a96f5683a62ebc7 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261106-r06-linear-diophantine-equation-0035.json)