# Natural GCD, Euclid Traces, and Coprimality

This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0001/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0002/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0003/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0004/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0005/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0006/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0007/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0008/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0009/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0010/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0011/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0012/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0013/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0014/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0015/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0016/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0017/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0018/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0019/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0020/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0021/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0022/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0023/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0024/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0025/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0026/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0027/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0028/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0029/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0030/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0031/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0032/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0033/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0034/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0035/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0036/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0037/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0038/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0039/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0040/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0041/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0042/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0043/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0044/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0045/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0046/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0047/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0048/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0049/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0050/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0051/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0052/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0053/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0054/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0055/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0056/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0057/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0058/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0059/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0060/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0061/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0062/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0063/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0064/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0065/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0066/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0067/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0068/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0069/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0070/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0071/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0072/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0073/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0074/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0075/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0076/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0077/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0078/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0079/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0080/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0081/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0082/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0083/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0084/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0085/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0086/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0087/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0088/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0089/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0090/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0091/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0092/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0093/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0094/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0095/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0096/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0097/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0098/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0099/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r03-gcd-coprime-0100/)

Greatest common divisor and coprimality · qf-ext18-gcd-associative-triple · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261013-advance-20261018-r03-gcd-coprime-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261013-advance-20261018-r03-gcd-coprime-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261013-advance-20261018-r03-gcd-coprime-foundations/validation-report.json)