Natural GCD, Euclid Traces, and Coprimality
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This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.
batch-20261013-advance-20261018-r03-gcd-coprime-foundations · 2026-10-13 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
1. Exact Euclidean Remainder Trace and Natural GCD
Use natural gcd with a=0, b=0. Give (a, b, ordered Euclid rows [x,y,x mod y] while y≠0, final gcd). Adopt gcd(0,0)=0.
View answer and explanation
(0, 0, [], 0)
Never take a remainder with zero divisor. Terminate at y=0 and retain the final x; coprimality does not mean both inputs are prime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (0, 0, [], 0).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-euclid-trace · candidate
2. Exact Euclidean Remainder Trace and Natural GCD
Use natural gcd with a=42, b=17. Give (a, b, ordered Euclid rows [x,y,x mod y] while y≠0, final gcd). Adopt gcd(0,0)=0.
View answer and explanation
(42, 17, [[42, 17, 8], [17, 8, 1], [8, 1, 0]], 1)
Never take a remainder with zero divisor. Terminate at y=0 and retain the final x; coprimality does not mean both inputs are prime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (42, 17, [[42, 17, 8], [17, 8, 1], [8, 1, 0]], 1).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-euclid-trace · candidate
3. Exact Euclidean Remainder Trace and Natural GCD
Use natural gcd with a=49, b=22. Give (a, b, ordered Euclid rows [x,y,x mod y] while y≠0, final gcd). Adopt gcd(0,0)=0.
View answer and explanation
(49, 22, [[49, 22, 5], [22, 5, 2], [5, 2, 1], [2, 1, 0]], 1)
Never take a remainder with zero divisor. Terminate at y=0 and retain the final x; coprimality does not mean both inputs are prime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (49, 22, [[49, 22, 5], [22, 5, 2], [5, 2, 1], [2, 1, 0]], 1).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-euclid-trace · candidate
4. Exact Euclidean Remainder Trace and Natural GCD
Use natural gcd with a=56, b=27. Give (a, b, ordered Euclid rows [x,y,x mod y] while y≠0, final gcd). Adopt gcd(0,0)=0.
View answer and explanation
(56, 27, [[56, 27, 2], [27, 2, 1], [2, 1, 0]], 1)
Never take a remainder with zero divisor. Terminate at y=0 and retain the final x; coprimality does not mean both inputs are prime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (56, 27, [[56, 27, 2], [27, 2, 1], [2, 1, 0]], 1).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-euclid-trace · candidate
5. Exact Euclidean Remainder Trace and Natural GCD
Use natural gcd with a=63, b=32. Give (a, b, ordered Euclid rows [x,y,x mod y] while y≠0, final gcd). Adopt gcd(0,0)=0.
View answer and explanation
(63, 32, [[63, 32, 31], [32, 31, 1], [31, 1, 0]], 1)
Never take a remainder with zero divisor. Terminate at y=0 and retain the final x; coprimality does not mean both inputs are prime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (63, 32, [[63, 32, 31], [32, 31, 1], [31, 1, 0]], 1).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-euclid-trace · candidate
6. Exact Euclidean Remainder Trace and Natural GCD
Use natural gcd with a=70, b=37. Give (a, b, ordered Euclid rows [x,y,x mod y] while y≠0, final gcd). Adopt gcd(0,0)=0.
View answer and explanation
(70, 37, [[70, 37, 33], [37, 33, 4], [33, 4, 1], [4, 1, 0]], 1)
Never take a remainder with zero divisor. Terminate at y=0 and retain the final x; coprimality does not mean both inputs are prime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (70, 37, [[70, 37, 33], [37, 33, 4], [33, 4, 1], [4, 1, 0]], 1).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-euclid-trace · candidate
7. Exact Euclidean Remainder Trace and Natural GCD
Use natural gcd with a=77, b=42. Give (a, b, ordered Euclid rows [x,y,x mod y] while y≠0, final gcd). Adopt gcd(0,0)=0.
View answer and explanation
(77, 42, [[77, 42, 35], [42, 35, 7], [35, 7, 0]], 7)
Never take a remainder with zero divisor. Terminate at y=0 and retain the final x; coprimality does not mean both inputs are prime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (77, 42, [[77, 42, 35], [42, 35, 7], [35, 7, 0]], 7).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-euclid-trace · candidate
8. Exact Euclidean Remainder Trace and Natural GCD
Use natural gcd with a=84, b=47. Give (a, b, ordered Euclid rows [x,y,x mod y] while y≠0, final gcd). Adopt gcd(0,0)=0.
View answer and explanation
(84, 47, [[84, 47, 37], [47, 37, 10], [37, 10, 7], [10, 7, 3], [7, 3, 1], [3, 1, 0]], 1)
Never take a remainder with zero divisor. Terminate at y=0 and retain the final x; coprimality does not mean both inputs are prime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (84, 47, [[84, 47, 37], [47, 37, 10], [37, 10, 7], [10, 7, 3], [7, 3, 1], [3, 1, 0]], 1).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-euclid-trace · candidate
9. Exact Euclidean Remainder Trace and Natural GCD
Use natural gcd with a=91, b=52. Give (a, b, ordered Euclid rows [x,y,x mod y] while y≠0, final gcd). Adopt gcd(0,0)=0.
View answer and explanation
(91, 52, [[91, 52, 39], [52, 39, 13], [39, 13, 0]], 13)
Never take a remainder with zero divisor. Terminate at y=0 and retain the final x; coprimality does not mean both inputs are prime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (91, 52, [[91, 52, 39], [52, 39, 13], [39, 13, 0]], 13).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-euclid-trace · candidate
10. Exact Euclidean Remainder Trace and Natural GCD
Use natural gcd with a=98, b=57. Give (a, b, ordered Euclid rows [x,y,x mod y] while y≠0, final gcd). Adopt gcd(0,0)=0.
View answer and explanation
(98, 57, [[98, 57, 41], [57, 41, 16], [41, 16, 9], [16, 9, 7], [9, 7, 2], [7, 2, 1], [2, 1, 0]], 1)
Never take a remainder with zero divisor. Terminate at y=0 and retain the final x; coprimality does not mean both inputs are prime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (98, 57, [[98, 57, 41], [57, 41, 16], [41, 16, 9], [16, 9, 7], [9, 7, 2], [7, 2, 1], [2, 1, 0]], 1).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-euclid-trace · candidate
11. Exact Euclidean Remainder Trace and Natural GCD
Use natural gcd with a=105, b=62. Give (a, b, ordered Euclid rows [x,y,x mod y] while y≠0, final gcd). Adopt gcd(0,0)=0.
View answer and explanation
(105, 62, [[105, 62, 43], [62, 43, 19], [43, 19, 5], [19, 5, 4], [5, 4, 1], [4, 1, 0]], 1)
Never take a remainder with zero divisor. Terminate at y=0 and retain the final x; coprimality does not mean both inputs are prime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (105, 62, [[105, 62, 43], [62, 43, 19], [43, 19, 5], [19, 5, 4], [5, 4, 1], [4, 1, 0]], 1).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-euclid-trace · candidate
12. Exact Euclidean Remainder Trace and Natural GCD
Use natural gcd with a=112, b=67. Give (a, b, ordered Euclid rows [x,y,x mod y] while y≠0, final gcd). Adopt gcd(0,0)=0.
View answer and explanation
(112, 67, [[112, 67, 45], [67, 45, 22], [45, 22, 1], [22, 1, 0]], 1)
Never take a remainder with zero divisor. Terminate at y=0 and retain the final x; coprimality does not mean both inputs are prime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (112, 67, [[112, 67, 45], [67, 45, 22], [45, 22, 1], [22, 1, 0]], 1).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-euclid-trace · candidate
13. Exact Euclidean Remainder Trace and Natural GCD
Use natural gcd with a=119, b=72. Give (a, b, ordered Euclid rows [x,y,x mod y] while y≠0, final gcd). Adopt gcd(0,0)=0.
View answer and explanation
(119, 72, [[119, 72, 47], [72, 47, 25], [47, 25, 22], [25, 22, 3], [22, 3, 1], [3, 1, 0]], 1)
Never take a remainder with zero divisor. Terminate at y=0 and retain the final x; coprimality does not mean both inputs are prime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (119, 72, [[119, 72, 47], [72, 47, 25], [47, 25, 22], [25, 22, 3], [22, 3, 1], [3, 1, 0]], 1).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-euclid-trace · candidate
14. Exact Euclidean Remainder Trace and Natural GCD
Use natural gcd with a=126, b=77. Give (a, b, ordered Euclid rows [x,y,x mod y] while y≠0, final gcd). Adopt gcd(0,0)=0.
View answer and explanation
(126, 77, [[126, 77, 49], [77, 49, 28], [49, 28, 21], [28, 21, 7], [21, 7, 0]], 7)
Never take a remainder with zero divisor. Terminate at y=0 and retain the final x; coprimality does not mean both inputs are prime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (126, 77, [[126, 77, 49], [77, 49, 28], [49, 28, 21], [28, 21, 7], [21, 7, 0]], 7).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-euclid-trace · candidate
15. Exact Euclidean Remainder Trace and Natural GCD
Use natural gcd with a=133, b=82. Give (a, b, ordered Euclid rows [x,y,x mod y] while y≠0, final gcd). Adopt gcd(0,0)=0.
View answer and explanation
(133, 82, [[133, 82, 51], [82, 51, 31], [51, 31, 20], [31, 20, 11], [20, 11, 9], [11, 9, 2], [9, 2, 1], [2, 1, 0]], 1)
Never take a remainder with zero divisor. Terminate at y=0 and retain the final x; coprimality does not mean both inputs are prime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (133, 82, [[133, 82, 51], [82, 51, 31], [51, 31, 20], [31, 20, 11], [20, 11, 9], [11, 9, 2], [9, 2, 1], [2, 1, 0]], 1).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-euclid-trace · candidate
16. Exact Euclidean Remainder Trace and Natural GCD
Use natural gcd with a=140, b=87. Give (a, b, ordered Euclid rows [x,y,x mod y] while y≠0, final gcd). Adopt gcd(0,0)=0.
View answer and explanation
(140, 87, [[140, 87, 53], [87, 53, 34], [53, 34, 19], [34, 19, 15], [19, 15, 4], [15, 4, 3], [4, 3, 1], [3, 1, 0]], 1)
Never take a remainder with zero divisor. Terminate at y=0 and retain the final x; coprimality does not mean both inputs are prime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (140, 87, [[140, 87, 53], [87, 53, 34], [53, 34, 19], [34, 19, 15], [19, 15, 4], [15, 4, 3], [4, 3, 1], [3, 1, 0]], 1).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-euclid-trace · candidate
17. Exact Euclidean Remainder Trace and Natural GCD
Use natural gcd with a=147, b=92. Give (a, b, ordered Euclid rows [x,y,x mod y] while y≠0, final gcd). Adopt gcd(0,0)=0.
View answer and explanation
(147, 92, [[147, 92, 55], [92, 55, 37], [55, 37, 18], [37, 18, 1], [18, 1, 0]], 1)
Never take a remainder with zero divisor. Terminate at y=0 and retain the final x; coprimality does not mean both inputs are prime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (147, 92, [[147, 92, 55], [92, 55, 37], [55, 37, 18], [37, 18, 1], [18, 1, 0]], 1).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-euclid-trace · candidate
18. Exact Euclidean Remainder Trace and Natural GCD
Use natural gcd with a=154, b=97. Give (a, b, ordered Euclid rows [x,y,x mod y] while y≠0, final gcd). Adopt gcd(0,0)=0.
View answer and explanation
(154, 97, [[154, 97, 57], [97, 57, 40], [57, 40, 17], [40, 17, 6], [17, 6, 5], [6, 5, 1], [5, 1, 0]], 1)
Never take a remainder with zero divisor. Terminate at y=0 and retain the final x; coprimality does not mean both inputs are prime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (154, 97, [[154, 97, 57], [97, 57, 40], [57, 40, 17], [40, 17, 6], [17, 6, 5], [6, 5, 1], [5, 1, 0]], 1).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-euclid-trace · candidate
19. Exact Euclidean Remainder Trace and Natural GCD
Use natural gcd with a=161, b=102. Give (a, b, ordered Euclid rows [x,y,x mod y] while y≠0, final gcd). Adopt gcd(0,0)=0.
View answer and explanation
(161, 102, [[161, 102, 59], [102, 59, 43], [59, 43, 16], [43, 16, 11], [16, 11, 5], [11, 5, 1], [5, 1, 0]], 1)
Never take a remainder with zero divisor. Terminate at y=0 and retain the final x; coprimality does not mean both inputs are prime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (161, 102, [[161, 102, 59], [102, 59, 43], [59, 43, 16], [43, 16, 11], [16, 11, 5], [11, 5, 1], [5, 1, 0]], 1).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-euclid-trace · candidate
20. Exact Euclidean Remainder Trace and Natural GCD
Use natural gcd with a=168, b=107. Give (a, b, ordered Euclid rows [x,y,x mod y] while y≠0, final gcd). Adopt gcd(0,0)=0.
View answer and explanation
(168, 107, [[168, 107, 61], [107, 61, 46], [61, 46, 15], [46, 15, 1], [15, 1, 0]], 1)
Never take a remainder with zero divisor. Terminate at y=0 and retain the final x; coprimality does not mean both inputs are prime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (168, 107, [[168, 107, 61], [107, 61, 46], [61, 46, 15], [46, 15, 1], [15, 1, 0]], 1).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-euclid-trace · candidate
21. Exact Euclidean Remainder Trace and Natural GCD
Use natural gcd with a=175, b=112. Give (a, b, ordered Euclid rows [x,y,x mod y] while y≠0, final gcd). Adopt gcd(0,0)=0.
View answer and explanation
(175, 112, [[175, 112, 63], [112, 63, 49], [63, 49, 14], [49, 14, 7], [14, 7, 0]], 7)
Never take a remainder with zero divisor. Terminate at y=0 and retain the final x; coprimality does not mean both inputs are prime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (175, 112, [[175, 112, 63], [112, 63, 49], [63, 49, 14], [49, 14, 7], [14, 7, 0]], 7).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-euclid-trace · candidate
22. Exact Euclidean Remainder Trace and Natural GCD
Use natural gcd with a=182, b=117. Give (a, b, ordered Euclid rows [x,y,x mod y] while y≠0, final gcd). Adopt gcd(0,0)=0.
View answer and explanation
(182, 117, [[182, 117, 65], [117, 65, 52], [65, 52, 13], [52, 13, 0]], 13)
Never take a remainder with zero divisor. Terminate at y=0 and retain the final x; coprimality does not mean both inputs are prime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (182, 117, [[182, 117, 65], [117, 65, 52], [65, 52, 13], [52, 13, 0]], 13).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-euclid-trace · candidate
23. Exact Euclidean Remainder Trace and Natural GCD
Use natural gcd with a=189, b=122. Give (a, b, ordered Euclid rows [x,y,x mod y] while y≠0, final gcd). Adopt gcd(0,0)=0.
View answer and explanation
(189, 122, [[189, 122, 67], [122, 67, 55], [67, 55, 12], [55, 12, 7], [12, 7, 5], [7, 5, 2], [5, 2, 1], [2, 1, 0]], 1)
Never take a remainder with zero divisor. Terminate at y=0 and retain the final x; coprimality does not mean both inputs are prime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (189, 122, [[189, 122, 67], [122, 67, 55], [67, 55, 12], [55, 12, 7], [12, 7, 5], [7, 5, 2], [5, 2, 1], [2, 1, 0]], 1).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-euclid-trace · candidate
24. Exact Euclidean Remainder Trace and Natural GCD
Use natural gcd with a=196, b=127. Give (a, b, ordered Euclid rows [x,y,x mod y] while y≠0, final gcd). Adopt gcd(0,0)=0.
View answer and explanation
(196, 127, [[196, 127, 69], [127, 69, 58], [69, 58, 11], [58, 11, 3], [11, 3, 2], [3, 2, 1], [2, 1, 0]], 1)
Never take a remainder with zero divisor. Terminate at y=0 and retain the final x; coprimality does not mean both inputs are prime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (196, 127, [[196, 127, 69], [127, 69, 58], [69, 58, 11], [58, 11, 3], [11, 3, 2], [3, 2, 1], [2, 1, 0]], 1).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-euclid-trace · candidate
25. Exact Euclidean Remainder Trace and Natural GCD
Use natural gcd with a=203, b=132. Give (a, b, ordered Euclid rows [x,y,x mod y] while y≠0, final gcd). Adopt gcd(0,0)=0.
View answer and explanation
(203, 132, [[203, 132, 71], [132, 71, 61], [71, 61, 10], [61, 10, 1], [10, 1, 0]], 1)
Never take a remainder with zero divisor. Terminate at y=0 and retain the final x; coprimality does not mean both inputs are prime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (203, 132, [[203, 132, 71], [132, 71, 61], [71, 61, 10], [61, 10, 1], [10, 1, 0]], 1).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-euclid-trace · candidate
26. A Common Divisor versus the Actual GCD
Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=36, b=24, candidate d=12; adopt gcd(0,0)=0. Give (a, b, d, d divides a?, d divides b?, d equals gcd?, gcd, both common-divisor and gcd qualifications?).
View answer and explanation
(36, 24, 12, Yes, Yes, Yes, 12, Yes)
Recompute gcd independently of the candidate. Dividing both inputs alone does not establish greatestness. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (36, 24, 12, Yes, Yes, Yes, 12, Yes).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-common-divisor-audit · candidate
27. A Common Divisor versus the Actual GCD
Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=40, b=30, candidate d=1; adopt gcd(0,0)=0. Give (a, b, d, d divides a?, d divides b?, d equals gcd?, gcd, both common-divisor and gcd qualifications?).
View answer and explanation
(40, 30, 1, Yes, Yes, No, 10, No)
Recompute gcd independently of the candidate. Dividing both inputs alone does not establish greatestness. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (40, 30, 1, Yes, Yes, No, 10, No).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-common-divisor-audit · candidate
28. A Common Divisor versus the Actual GCD
Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=44, b=36, candidate d=5; adopt gcd(0,0)=0. Give (a, b, d, d divides a?, d divides b?, d equals gcd?, gcd, both common-divisor and gcd qualifications?).
View answer and explanation
(44, 36, 5, No, No, No, 4, No)
Recompute gcd independently of the candidate. Dividing both inputs alone does not establish greatestness. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (44, 36, 5, No, No, No, 4, No).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-common-divisor-audit · candidate
29. A Common Divisor versus the Actual GCD
Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=48, b=42, candidate d=6; adopt gcd(0,0)=0. Give (a, b, d, d divides a?, d divides b?, d equals gcd?, gcd, both common-divisor and gcd qualifications?).
View answer and explanation
(48, 42, 6, Yes, Yes, Yes, 6, Yes)
Recompute gcd independently of the candidate. Dividing both inputs alone does not establish greatestness. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (48, 42, 6, Yes, Yes, Yes, 6, Yes).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-common-divisor-audit · candidate
30. A Common Divisor versus the Actual GCD
Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=52, b=48, candidate d=1; adopt gcd(0,0)=0. Give (a, b, d, d divides a?, d divides b?, d equals gcd?, gcd, both common-divisor and gcd qualifications?).
View answer and explanation
(52, 48, 1, Yes, Yes, No, 4, No)
Recompute gcd independently of the candidate. Dividing both inputs alone does not establish greatestness. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (52, 48, 1, Yes, Yes, No, 4, No).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-common-divisor-audit · candidate
31. A Common Divisor versus the Actual GCD
Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=56, b=54, candidate d=3; adopt gcd(0,0)=0. Give (a, b, d, d divides a?, d divides b?, d equals gcd?, gcd, both common-divisor and gcd qualifications?).
View answer and explanation
(56, 54, 3, No, Yes, No, 2, No)
Recompute gcd independently of the candidate. Dividing both inputs alone does not establish greatestness. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (56, 54, 3, No, Yes, No, 2, No).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-common-divisor-audit · candidate
32. A Common Divisor versus the Actual GCD
Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=60, b=60, candidate d=60; adopt gcd(0,0)=0. Give (a, b, d, d divides a?, d divides b?, d equals gcd?, gcd, both common-divisor and gcd qualifications?).
View answer and explanation
(60, 60, 60, Yes, Yes, Yes, 60, Yes)
Recompute gcd independently of the candidate. Dividing both inputs alone does not establish greatestness. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (60, 60, 60, Yes, Yes, Yes, 60, Yes).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-common-divisor-audit · candidate
33. A Common Divisor versus the Actual GCD
Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=64, b=66, candidate d=1; adopt gcd(0,0)=0. Give (a, b, d, d divides a?, d divides b?, d equals gcd?, gcd, both common-divisor and gcd qualifications?).
View answer and explanation
(64, 66, 1, Yes, Yes, No, 2, No)
Recompute gcd independently of the candidate. Dividing both inputs alone does not establish greatestness. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (64, 66, 1, Yes, Yes, No, 2, No).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-common-divisor-audit · candidate
34. A Common Divisor versus the Actual GCD
Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=68, b=72, candidate d=5; adopt gcd(0,0)=0. Give (a, b, d, d divides a?, d divides b?, d equals gcd?, gcd, both common-divisor and gcd qualifications?).
View answer and explanation
(68, 72, 5, No, No, No, 4, No)
Recompute gcd independently of the candidate. Dividing both inputs alone does not establish greatestness. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (68, 72, 5, No, No, No, 4, No).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-common-divisor-audit · candidate
35. A Common Divisor versus the Actual GCD
Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=72, b=78, candidate d=6; adopt gcd(0,0)=0. Give (a, b, d, d divides a?, d divides b?, d equals gcd?, gcd, both common-divisor and gcd qualifications?).
View answer and explanation
(72, 78, 6, Yes, Yes, Yes, 6, Yes)
Recompute gcd independently of the candidate. Dividing both inputs alone does not establish greatestness. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (72, 78, 6, Yes, Yes, Yes, 6, Yes).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-common-divisor-audit · candidate
36. A Common Divisor versus the Actual GCD
Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=76, b=84, candidate d=1; adopt gcd(0,0)=0. Give (a, b, d, d divides a?, d divides b?, d equals gcd?, gcd, both common-divisor and gcd qualifications?).
View answer and explanation
(76, 84, 1, Yes, Yes, No, 4, No)
Recompute gcd independently of the candidate. Dividing both inputs alone does not establish greatestness. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (76, 84, 1, Yes, Yes, No, 4, No).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-common-divisor-audit · candidate
37. A Common Divisor versus the Actual GCD
Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=80, b=90, candidate d=11; adopt gcd(0,0)=0. Give (a, b, d, d divides a?, d divides b?, d equals gcd?, gcd, both common-divisor and gcd qualifications?).
View answer and explanation
(80, 90, 11, No, No, No, 10, No)
Recompute gcd independently of the candidate. Dividing both inputs alone does not establish greatestness. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (80, 90, 11, No, No, No, 10, No).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-common-divisor-audit · candidate
38. A Common Divisor versus the Actual GCD
Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=84, b=96, candidate d=12; adopt gcd(0,0)=0. Give (a, b, d, d divides a?, d divides b?, d equals gcd?, gcd, both common-divisor and gcd qualifications?).
View answer and explanation
(84, 96, 12, Yes, Yes, Yes, 12, Yes)
Recompute gcd independently of the candidate. Dividing both inputs alone does not establish greatestness. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (84, 96, 12, Yes, Yes, Yes, 12, Yes).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-common-divisor-audit · candidate
39. A Common Divisor versus the Actual GCD
Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=88, b=102, candidate d=1; adopt gcd(0,0)=0. Give (a, b, d, d divides a?, d divides b?, d equals gcd?, gcd, both common-divisor and gcd qualifications?).
View answer and explanation
(88, 102, 1, Yes, Yes, No, 2, No)
Recompute gcd independently of the candidate. Dividing both inputs alone does not establish greatestness. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (88, 102, 1, Yes, Yes, No, 2, No).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-common-divisor-audit · candidate
40. A Common Divisor versus the Actual GCD
Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=92, b=108, candidate d=5; adopt gcd(0,0)=0. Give (a, b, d, d divides a?, d divides b?, d equals gcd?, gcd, both common-divisor and gcd qualifications?).
View answer and explanation
(92, 108, 5, No, No, No, 4, No)
Recompute gcd independently of the candidate. Dividing both inputs alone does not establish greatestness. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (92, 108, 5, No, No, No, 4, No).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-common-divisor-audit · candidate
41. A Common Divisor versus the Actual GCD
Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=96, b=114, candidate d=6; adopt gcd(0,0)=0. Give (a, b, d, d divides a?, d divides b?, d equals gcd?, gcd, both common-divisor and gcd qualifications?).
View answer and explanation
(96, 114, 6, Yes, Yes, Yes, 6, Yes)
Recompute gcd independently of the candidate. Dividing both inputs alone does not establish greatestness. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (96, 114, 6, Yes, Yes, Yes, 6, Yes).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-common-divisor-audit · candidate
42. A Common Divisor versus the Actual GCD
Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=100, b=120, candidate d=1; adopt gcd(0,0)=0. Give (a, b, d, d divides a?, d divides b?, d equals gcd?, gcd, both common-divisor and gcd qualifications?).
View answer and explanation
(100, 120, 1, Yes, Yes, No, 20, No)
Recompute gcd independently of the candidate. Dividing both inputs alone does not establish greatestness. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (100, 120, 1, Yes, Yes, No, 20, No).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-common-divisor-audit · candidate
43. A Common Divisor versus the Actual GCD
Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=104, b=126, candidate d=3; adopt gcd(0,0)=0. Give (a, b, d, d divides a?, d divides b?, d equals gcd?, gcd, both common-divisor and gcd qualifications?).
View answer and explanation
(104, 126, 3, No, Yes, No, 2, No)
Recompute gcd independently of the candidate. Dividing both inputs alone does not establish greatestness. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (104, 126, 3, No, Yes, No, 2, No).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-common-divisor-audit · candidate
44. A Common Divisor versus the Actual GCD
Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=108, b=132, candidate d=12; adopt gcd(0,0)=0. Give (a, b, d, d divides a?, d divides b?, d equals gcd?, gcd, both common-divisor and gcd qualifications?).
View answer and explanation
(108, 132, 12, Yes, Yes, Yes, 12, Yes)
Recompute gcd independently of the candidate. Dividing both inputs alone does not establish greatestness. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (108, 132, 12, Yes, Yes, Yes, 12, Yes).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-common-divisor-audit · candidate
45. A Common Divisor versus the Actual GCD
Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=112, b=138, candidate d=1; adopt gcd(0,0)=0. Give (a, b, d, d divides a?, d divides b?, d equals gcd?, gcd, both common-divisor and gcd qualifications?).
View answer and explanation
(112, 138, 1, Yes, Yes, No, 2, No)
Recompute gcd independently of the candidate. Dividing both inputs alone does not establish greatestness. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (112, 138, 1, Yes, Yes, No, 2, No).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-common-divisor-audit · candidate
46. A Common Divisor versus the Actual GCD
Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=116, b=144, candidate d=5; adopt gcd(0,0)=0. Give (a, b, d, d divides a?, d divides b?, d equals gcd?, gcd, both common-divisor and gcd qualifications?).
View answer and explanation
(116, 144, 5, No, No, No, 4, No)
Recompute gcd independently of the candidate. Dividing both inputs alone does not establish greatestness. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (116, 144, 5, No, No, No, 4, No).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-common-divisor-audit · candidate
47. A Common Divisor versus the Actual GCD
Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=120, b=150, candidate d=30; adopt gcd(0,0)=0. Give (a, b, d, d divides a?, d divides b?, d equals gcd?, gcd, both common-divisor and gcd qualifications?).
View answer and explanation
(120, 150, 30, Yes, Yes, Yes, 30, Yes)
Recompute gcd independently of the candidate. Dividing both inputs alone does not establish greatestness. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (120, 150, 30, Yes, Yes, Yes, 30, Yes).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-common-divisor-audit · candidate
48. A Common Divisor versus the Actual GCD
Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=124, b=156, candidate d=1; adopt gcd(0,0)=0. Give (a, b, d, d divides a?, d divides b?, d equals gcd?, gcd, both common-divisor and gcd qualifications?).
View answer and explanation
(124, 156, 1, Yes, Yes, No, 4, No)
Recompute gcd independently of the candidate. Dividing both inputs alone does not establish greatestness. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (124, 156, 1, Yes, Yes, No, 4, No).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-common-divisor-audit · candidate
49. A Common Divisor versus the Actual GCD
Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=128, b=162, candidate d=3; adopt gcd(0,0)=0. Give (a, b, d, d divides a?, d divides b?, d equals gcd?, gcd, both common-divisor and gcd qualifications?).
View answer and explanation
(128, 162, 3, No, Yes, No, 2, No)
Recompute gcd independently of the candidate. Dividing both inputs alone does not establish greatestness. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (128, 162, 3, No, Yes, No, 2, No).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-common-divisor-audit · candidate
50. A Common Divisor versus the Actual GCD
Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. a=132, b=168, candidate d=12; adopt gcd(0,0)=0. Give (a, b, d, d divides a?, d divides b?, d equals gcd?, gcd, both common-divisor and gcd qualifications?).
View answer and explanation
(132, 168, 12, Yes, Yes, Yes, 12, Yes)
Recompute gcd independently of the candidate. Dividing both inputs alone does not establish greatestness. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (132, 168, 12, Yes, Yes, Yes, 12, Yes).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-common-divisor-audit · candidate
51. Natural GCD and Coprimality Including Zero
a=0, b=1 are natural numbers, including 0; gcd(0,0)=0. Give (a, b, gcd, gcd=1?). Coprime does not mean that both inputs are prime.
View answer and explanation
(0, 1, 1, Yes)
Compute the exact gcd and compare it with 1; in particular, 0 and 1 are coprime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (0, 1, 1, Yes).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-coprime-classification · candidate
52. Natural GCD and Coprimality Including Zero
a=1, b=3 are natural numbers, including 0; gcd(0,0)=0. Give (a, b, gcd, gcd=1?). Coprime does not mean that both inputs are prime.
View answer and explanation
(1, 3, 1, Yes)
Compute the exact gcd and compare it with 1; in particular, 0 and 1 are coprime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (1, 3, 1, Yes).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-coprime-classification · candidate
53. Natural GCD and Coprimality Including Zero
a=2, b=5 are natural numbers, including 0; gcd(0,0)=0. Give (a, b, gcd, gcd=1?). Coprime does not mean that both inputs are prime.
View answer and explanation
(2, 5, 1, Yes)
Compute the exact gcd and compare it with 1; in particular, 0 and 1 are coprime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (2, 5, 1, Yes).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-coprime-classification · candidate
54. Natural GCD and Coprimality Including Zero
a=3, b=7 are natural numbers, including 0; gcd(0,0)=0. Give (a, b, gcd, gcd=1?). Coprime does not mean that both inputs are prime.
View answer and explanation
(3, 7, 1, Yes)
Compute the exact gcd and compare it with 1; in particular, 0 and 1 are coprime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, 7, 1, Yes).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-coprime-classification · candidate
55. Natural GCD and Coprimality Including Zero
a=4, b=9 are natural numbers, including 0; gcd(0,0)=0. Give (a, b, gcd, gcd=1?). Coprime does not mean that both inputs are prime.
View answer and explanation
(4, 9, 1, Yes)
Compute the exact gcd and compare it with 1; in particular, 0 and 1 are coprime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 9, 1, Yes).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-coprime-classification · candidate
56. Natural GCD and Coprimality Including Zero
a=5, b=11 are natural numbers, including 0; gcd(0,0)=0. Give (a, b, gcd, gcd=1?). Coprime does not mean that both inputs are prime.
View answer and explanation
(5, 11, 1, Yes)
Compute the exact gcd and compare it with 1; in particular, 0 and 1 are coprime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, 11, 1, Yes).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-coprime-classification · candidate
57. Natural GCD and Coprimality Including Zero
a=6, b=13 are natural numbers, including 0; gcd(0,0)=0. Give (a, b, gcd, gcd=1?). Coprime does not mean that both inputs are prime.
View answer and explanation
(6, 13, 1, Yes)
Compute the exact gcd and compare it with 1; in particular, 0 and 1 are coprime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, 13, 1, Yes).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-coprime-classification · candidate
58. Natural GCD and Coprimality Including Zero
a=7, b=15 are natural numbers, including 0; gcd(0,0)=0. Give (a, b, gcd, gcd=1?). Coprime does not mean that both inputs are prime.
View answer and explanation
(7, 15, 1, Yes)
Compute the exact gcd and compare it with 1; in particular, 0 and 1 are coprime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (7, 15, 1, Yes).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-coprime-classification · candidate
59. Natural GCD and Coprimality Including Zero
a=8, b=17 are natural numbers, including 0; gcd(0,0)=0. Give (a, b, gcd, gcd=1?). Coprime does not mean that both inputs are prime.
View answer and explanation
(8, 17, 1, Yes)
Compute the exact gcd and compare it with 1; in particular, 0 and 1 are coprime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (8, 17, 1, Yes).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-coprime-classification · candidate
60. Natural GCD and Coprimality Including Zero
a=9, b=19 are natural numbers, including 0; gcd(0,0)=0. Give (a, b, gcd, gcd=1?). Coprime does not mean that both inputs are prime.
View answer and explanation
(9, 19, 1, Yes)
Compute the exact gcd and compare it with 1; in particular, 0 and 1 are coprime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (9, 19, 1, Yes).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-coprime-classification · candidate
61. Natural GCD and Coprimality Including Zero
a=10, b=21 are natural numbers, including 0; gcd(0,0)=0. Give (a, b, gcd, gcd=1?). Coprime does not mean that both inputs are prime.
View answer and explanation
(10, 21, 1, Yes)
Compute the exact gcd and compare it with 1; in particular, 0 and 1 are coprime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (10, 21, 1, Yes).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-coprime-classification · candidate
62. Natural GCD and Coprimality Including Zero
a=11, b=23 are natural numbers, including 0; gcd(0,0)=0. Give (a, b, gcd, gcd=1?). Coprime does not mean that both inputs are prime.
View answer and explanation
(11, 23, 1, Yes)
Compute the exact gcd and compare it with 1; in particular, 0 and 1 are coprime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (11, 23, 1, Yes).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-coprime-classification · candidate
63. Natural GCD and Coprimality Including Zero
a=12, b=25 are natural numbers, including 0; gcd(0,0)=0. Give (a, b, gcd, gcd=1?). Coprime does not mean that both inputs are prime.
View answer and explanation
(12, 25, 1, Yes)
Compute the exact gcd and compare it with 1; in particular, 0 and 1 are coprime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (12, 25, 1, Yes).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-coprime-classification · candidate
64. Natural GCD and Coprimality Including Zero
a=13, b=27 are natural numbers, including 0; gcd(0,0)=0. Give (a, b, gcd, gcd=1?). Coprime does not mean that both inputs are prime.
View answer and explanation
(13, 27, 1, Yes)
Compute the exact gcd and compare it with 1; in particular, 0 and 1 are coprime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (13, 27, 1, Yes).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-coprime-classification · candidate
65. Natural GCD and Coprimality Including Zero
a=14, b=29 are natural numbers, including 0; gcd(0,0)=0. Give (a, b, gcd, gcd=1?). Coprime does not mean that both inputs are prime.
View answer and explanation
(14, 29, 1, Yes)
Compute the exact gcd and compare it with 1; in particular, 0 and 1 are coprime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (14, 29, 1, Yes).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-coprime-classification · candidate
66. Natural GCD and Coprimality Including Zero
a=15, b=31 are natural numbers, including 0; gcd(0,0)=0. Give (a, b, gcd, gcd=1?). Coprime does not mean that both inputs are prime.
View answer and explanation
(15, 31, 1, Yes)
Compute the exact gcd and compare it with 1; in particular, 0 and 1 are coprime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (15, 31, 1, Yes).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-coprime-classification · candidate
67. Natural GCD and Coprimality Including Zero
a=16, b=33 are natural numbers, including 0; gcd(0,0)=0. Give (a, b, gcd, gcd=1?). Coprime does not mean that both inputs are prime.
View answer and explanation
(16, 33, 1, Yes)
Compute the exact gcd and compare it with 1; in particular, 0 and 1 are coprime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (16, 33, 1, Yes).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-coprime-classification · candidate
68. Natural GCD and Coprimality Including Zero
a=17, b=35 are natural numbers, including 0; gcd(0,0)=0. Give (a, b, gcd, gcd=1?). Coprime does not mean that both inputs are prime.
View answer and explanation
(17, 35, 1, Yes)
Compute the exact gcd and compare it with 1; in particular, 0 and 1 are coprime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (17, 35, 1, Yes).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-coprime-classification · candidate
69. Natural GCD and Coprimality Including Zero
a=18, b=37 are natural numbers, including 0; gcd(0,0)=0. Give (a, b, gcd, gcd=1?). Coprime does not mean that both inputs are prime.
View answer and explanation
(18, 37, 1, Yes)
Compute the exact gcd and compare it with 1; in particular, 0 and 1 are coprime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (18, 37, 1, Yes).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-coprime-classification · candidate
70. Natural GCD and Coprimality Including Zero
a=19, b=39 are natural numbers, including 0; gcd(0,0)=0. Give (a, b, gcd, gcd=1?). Coprime does not mean that both inputs are prime.
View answer and explanation
(19, 39, 1, Yes)
Compute the exact gcd and compare it with 1; in particular, 0 and 1 are coprime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (19, 39, 1, Yes).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-coprime-classification · candidate
71. Natural GCD and Coprimality Including Zero
a=20, b=41 are natural numbers, including 0; gcd(0,0)=0. Give (a, b, gcd, gcd=1?). Coprime does not mean that both inputs are prime.
View answer and explanation
(20, 41, 1, Yes)
Compute the exact gcd and compare it with 1; in particular, 0 and 1 are coprime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (20, 41, 1, Yes).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-coprime-classification · candidate
72. Natural GCD and Coprimality Including Zero
a=21, b=43 are natural numbers, including 0; gcd(0,0)=0. Give (a, b, gcd, gcd=1?). Coprime does not mean that both inputs are prime.
View answer and explanation
(21, 43, 1, Yes)
Compute the exact gcd and compare it with 1; in particular, 0 and 1 are coprime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (21, 43, 1, Yes).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-coprime-classification · candidate
73. Natural GCD and Coprimality Including Zero
a=22, b=45 are natural numbers, including 0; gcd(0,0)=0. Give (a, b, gcd, gcd=1?). Coprime does not mean that both inputs are prime.
View answer and explanation
(22, 45, 1, Yes)
Compute the exact gcd and compare it with 1; in particular, 0 and 1 are coprime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (22, 45, 1, Yes).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-coprime-classification · candidate
74. Natural GCD and Coprimality Including Zero
a=23, b=47 are natural numbers, including 0; gcd(0,0)=0. Give (a, b, gcd, gcd=1?). Coprime does not mean that both inputs are prime.
View answer and explanation
(23, 47, 1, Yes)
Compute the exact gcd and compare it with 1; in particular, 0 and 1 are coprime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (23, 47, 1, Yes).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-coprime-classification · candidate
75. Natural GCD and Coprimality Including Zero
a=24, b=49 are natural numbers, including 0; gcd(0,0)=0. Give (a, b, gcd, gcd=1?). Coprime does not mean that both inputs are prime.
View answer and explanation
(24, 49, 1, Yes)
Compute the exact gcd and compare it with 1; in particular, 0 and 1 are coprime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (24, 49, 1, Yes).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-coprime-classification · candidate
76. Both Parenthesizations of a Three-Input GCD
Natural inputs a=12, b=18, c=30, with gcd(0,0)=0. Give (a, b, c, gcd(gcd(a,b),c), gcd(a,gcd(b,c)), are the results equal?).
View answer and explanation
(12, 18, 30, 6, 6, Yes)
Compute both complete gcd routes rather than merely repeating an asserted associativity law. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (12, 18, 30, 6, 6, Yes).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-associative-triple · candidate
77. Both Parenthesizations of a Three-Input GCD
Natural inputs a=13, b=20, c=33, with gcd(0,0)=0. Give (a, b, c, gcd(gcd(a,b),c), gcd(a,gcd(b,c)), are the results equal?).
View answer and explanation
(13, 20, 33, 1, 1, Yes)
Compute both complete gcd routes rather than merely repeating an asserted associativity law. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (13, 20, 33, 1, 1, Yes).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-associative-triple · candidate
78. Both Parenthesizations of a Three-Input GCD
Natural inputs a=14, b=22, c=36, with gcd(0,0)=0. Give (a, b, c, gcd(gcd(a,b),c), gcd(a,gcd(b,c)), are the results equal?).
View answer and explanation
(14, 22, 36, 2, 2, Yes)
Compute both complete gcd routes rather than merely repeating an asserted associativity law. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (14, 22, 36, 2, 2, Yes).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-associative-triple · candidate
79. Both Parenthesizations of a Three-Input GCD
Natural inputs a=15, b=24, c=39, with gcd(0,0)=0. Give (a, b, c, gcd(gcd(a,b),c), gcd(a,gcd(b,c)), are the results equal?).
View answer and explanation
(15, 24, 39, 3, 3, Yes)
Compute both complete gcd routes rather than merely repeating an asserted associativity law. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (15, 24, 39, 3, 3, Yes).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-associative-triple · candidate
80. Both Parenthesizations of a Three-Input GCD
Natural inputs a=16, b=26, c=42, with gcd(0,0)=0. Give (a, b, c, gcd(gcd(a,b),c), gcd(a,gcd(b,c)), are the results equal?).
View answer and explanation
(16, 26, 42, 2, 2, Yes)
Compute both complete gcd routes rather than merely repeating an asserted associativity law. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (16, 26, 42, 2, 2, Yes).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-associative-triple · candidate
81. Both Parenthesizations of a Three-Input GCD
Natural inputs a=17, b=28, c=45, with gcd(0,0)=0. Give (a, b, c, gcd(gcd(a,b),c), gcd(a,gcd(b,c)), are the results equal?).
View answer and explanation
(17, 28, 45, 1, 1, Yes)
Compute both complete gcd routes rather than merely repeating an asserted associativity law. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (17, 28, 45, 1, 1, Yes).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-associative-triple · candidate
82. Both Parenthesizations of a Three-Input GCD
Natural inputs a=18, b=30, c=48, with gcd(0,0)=0. Give (a, b, c, gcd(gcd(a,b),c), gcd(a,gcd(b,c)), are the results equal?).
View answer and explanation
(18, 30, 48, 6, 6, Yes)
Compute both complete gcd routes rather than merely repeating an asserted associativity law. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (18, 30, 48, 6, 6, Yes).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-associative-triple · candidate
83. Both Parenthesizations of a Three-Input GCD
Natural inputs a=19, b=32, c=51, with gcd(0,0)=0. Give (a, b, c, gcd(gcd(a,b),c), gcd(a,gcd(b,c)), are the results equal?).
View answer and explanation
(19, 32, 51, 1, 1, Yes)
Compute both complete gcd routes rather than merely repeating an asserted associativity law. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (19, 32, 51, 1, 1, Yes).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-associative-triple · candidate
84. Both Parenthesizations of a Three-Input GCD
Natural inputs a=20, b=34, c=54, with gcd(0,0)=0. Give (a, b, c, gcd(gcd(a,b),c), gcd(a,gcd(b,c)), are the results equal?).
View answer and explanation
(20, 34, 54, 2, 2, Yes)
Compute both complete gcd routes rather than merely repeating an asserted associativity law. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (20, 34, 54, 2, 2, Yes).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-associative-triple · candidate
85. Both Parenthesizations of a Three-Input GCD
Natural inputs a=21, b=36, c=57, with gcd(0,0)=0. Give (a, b, c, gcd(gcd(a,b),c), gcd(a,gcd(b,c)), are the results equal?).
View answer and explanation
(21, 36, 57, 3, 3, Yes)
Compute both complete gcd routes rather than merely repeating an asserted associativity law. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (21, 36, 57, 3, 3, Yes).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-associative-triple · candidate
86. Both Parenthesizations of a Three-Input GCD
Natural inputs a=22, b=38, c=60, with gcd(0,0)=0. Give (a, b, c, gcd(gcd(a,b),c), gcd(a,gcd(b,c)), are the results equal?).
View answer and explanation
(22, 38, 60, 2, 2, Yes)
Compute both complete gcd routes rather than merely repeating an asserted associativity law. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (22, 38, 60, 2, 2, Yes).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-associative-triple · candidate
87. Both Parenthesizations of a Three-Input GCD
Natural inputs a=23, b=40, c=63, with gcd(0,0)=0. Give (a, b, c, gcd(gcd(a,b),c), gcd(a,gcd(b,c)), are the results equal?).
View answer and explanation
(23, 40, 63, 1, 1, Yes)
Compute both complete gcd routes rather than merely repeating an asserted associativity law. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (23, 40, 63, 1, 1, Yes).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-associative-triple · candidate
88. Both Parenthesizations of a Three-Input GCD
Natural inputs a=24, b=42, c=66, with gcd(0,0)=0. Give (a, b, c, gcd(gcd(a,b),c), gcd(a,gcd(b,c)), are the results equal?).
View answer and explanation
(24, 42, 66, 6, 6, Yes)
Compute both complete gcd routes rather than merely repeating an asserted associativity law. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (24, 42, 66, 6, 6, Yes).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-associative-triple · candidate
89. Both Parenthesizations of a Three-Input GCD
Natural inputs a=25, b=44, c=69, with gcd(0,0)=0. Give (a, b, c, gcd(gcd(a,b),c), gcd(a,gcd(b,c)), are the results equal?).
View answer and explanation
(25, 44, 69, 1, 1, Yes)
Compute both complete gcd routes rather than merely repeating an asserted associativity law. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (25, 44, 69, 1, 1, Yes).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-associative-triple · candidate
90. Both Parenthesizations of a Three-Input GCD
Natural inputs a=26, b=46, c=72, with gcd(0,0)=0. Give (a, b, c, gcd(gcd(a,b),c), gcd(a,gcd(b,c)), are the results equal?).
View answer and explanation
(26, 46, 72, 2, 2, Yes)
Compute both complete gcd routes rather than merely repeating an asserted associativity law. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (26, 46, 72, 2, 2, Yes).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-associative-triple · candidate
91. Both Parenthesizations of a Three-Input GCD
Natural inputs a=27, b=48, c=75, with gcd(0,0)=0. Give (a, b, c, gcd(gcd(a,b),c), gcd(a,gcd(b,c)), are the results equal?).
View answer and explanation
(27, 48, 75, 3, 3, Yes)
Compute both complete gcd routes rather than merely repeating an asserted associativity law. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (27, 48, 75, 3, 3, Yes).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-associative-triple · candidate
92. Both Parenthesizations of a Three-Input GCD
Natural inputs a=28, b=50, c=78, with gcd(0,0)=0. Give (a, b, c, gcd(gcd(a,b),c), gcd(a,gcd(b,c)), are the results equal?).
View answer and explanation
(28, 50, 78, 2, 2, Yes)
Compute both complete gcd routes rather than merely repeating an asserted associativity law. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (28, 50, 78, 2, 2, Yes).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-associative-triple · candidate
93. Both Parenthesizations of a Three-Input GCD
Natural inputs a=29, b=52, c=81, with gcd(0,0)=0. Give (a, b, c, gcd(gcd(a,b),c), gcd(a,gcd(b,c)), are the results equal?).
View answer and explanation
(29, 52, 81, 1, 1, Yes)
Compute both complete gcd routes rather than merely repeating an asserted associativity law. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (29, 52, 81, 1, 1, Yes).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-associative-triple · candidate
94. Both Parenthesizations of a Three-Input GCD
Natural inputs a=30, b=54, c=84, with gcd(0,0)=0. Give (a, b, c, gcd(gcd(a,b),c), gcd(a,gcd(b,c)), are the results equal?).
View answer and explanation
(30, 54, 84, 6, 6, Yes)
Compute both complete gcd routes rather than merely repeating an asserted associativity law. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (30, 54, 84, 6, 6, Yes).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-associative-triple · candidate
95. Both Parenthesizations of a Three-Input GCD
Natural inputs a=31, b=56, c=87, with gcd(0,0)=0. Give (a, b, c, gcd(gcd(a,b),c), gcd(a,gcd(b,c)), are the results equal?).
View answer and explanation
(31, 56, 87, 1, 1, Yes)
Compute both complete gcd routes rather than merely repeating an asserted associativity law. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (31, 56, 87, 1, 1, Yes).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-associative-triple · candidate
96. Both Parenthesizations of a Three-Input GCD
Natural inputs a=32, b=58, c=90, with gcd(0,0)=0. Give (a, b, c, gcd(gcd(a,b),c), gcd(a,gcd(b,c)), are the results equal?).
View answer and explanation
(32, 58, 90, 2, 2, Yes)
Compute both complete gcd routes rather than merely repeating an asserted associativity law. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (32, 58, 90, 2, 2, Yes).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-associative-triple · candidate
97. Both Parenthesizations of a Three-Input GCD
Natural inputs a=33, b=60, c=93, with gcd(0,0)=0. Give (a, b, c, gcd(gcd(a,b),c), gcd(a,gcd(b,c)), are the results equal?).
View answer and explanation
(33, 60, 93, 3, 3, Yes)
Compute both complete gcd routes rather than merely repeating an asserted associativity law. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (33, 60, 93, 3, 3, Yes).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-associative-triple · candidate
98. Both Parenthesizations of a Three-Input GCD
Natural inputs a=34, b=62, c=96, with gcd(0,0)=0. Give (a, b, c, gcd(gcd(a,b),c), gcd(a,gcd(b,c)), are the results equal?).
View answer and explanation
(34, 62, 96, 2, 2, Yes)
Compute both complete gcd routes rather than merely repeating an asserted associativity law. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (34, 62, 96, 2, 2, Yes).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-associative-triple · candidate
99. Both Parenthesizations of a Three-Input GCD
Natural inputs a=35, b=64, c=99, with gcd(0,0)=0. Give (a, b, c, gcd(gcd(a,b),c), gcd(a,gcd(b,c)), are the results equal?).
View answer and explanation
(35, 64, 99, 1, 1, Yes)
Compute both complete gcd routes rather than merely repeating an asserted associativity law. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (35, 64, 99, 1, 1, Yes).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-associative-triple · candidate
100. Both Parenthesizations of a Three-Input GCD
Natural inputs a=36, b=66, c=102, with gcd(0,0)=0. Give (a, b, c, gcd(gcd(a,b),c), gcd(a,gcd(b,c)), are the results equal?).
View answer and explanation
(36, 66, 102, 6, 6, Yes)
Compute both complete gcd routes rather than merely repeating an asserted associativity law. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (36, 66, 102, 6, 6, Yes).
Reviewed step-by-step derivation
Greatest common divisor and coprimality · qf-ext18-gcd-associative-triple · candidate
Canonical question JSONL · English JSONL · Validation report