Positive-Modulus Natural Congruence and Guarded Preservation
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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-r05-modular-congruence-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. Positive-Modulus Guard before Natural Remainder Comparison
Fix modulus n=0, natural a=7, b=12. Give (n, a, b, n>0?, [a mod n,b mod n] or not-evaluated, congruent? or not-evaluated). This entry excludes zero-modulus conventions.
View answer and explanation
(0, 7, 12, No, not-evaluated, not-evaluated)
Check positivity before either remainder; no unconditional cancellation of common factors is permitted. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (0, 7, 12, No, not-evaluated, not-evaluated).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-remainder-gate · candidate
2. Positive-Modulus Guard before Natural Remainder Comparison
Fix modulus n=1, natural a=10, b=17. Give (n, a, b, n>0?, [a mod n,b mod n] or not-evaluated, congruent? or not-evaluated). This entry excludes zero-modulus conventions.
View answer and explanation
(1, 10, 17, Yes, [0, 0], Yes)
Check positivity before either remainder; no unconditional cancellation of common factors is permitted. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (1, 10, 17, Yes, [0, 0], Yes).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-remainder-gate · candidate
3. Positive-Modulus Guard before Natural Remainder Comparison
Fix modulus n=2, natural a=13, b=22. Give (n, a, b, n>0?, [a mod n,b mod n] or not-evaluated, congruent? or not-evaluated). This entry excludes zero-modulus conventions.
View answer and explanation
(2, 13, 22, Yes, [1, 0], No)
Check positivity before either remainder; no unconditional cancellation of common factors is permitted. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (2, 13, 22, Yes, [1, 0], No).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-remainder-gate · candidate
4. Positive-Modulus Guard before Natural Remainder Comparison
Fix modulus n=3, natural a=16, b=27. Give (n, a, b, n>0?, [a mod n,b mod n] or not-evaluated, congruent? or not-evaluated). This entry excludes zero-modulus conventions.
View answer and explanation
(3, 16, 27, Yes, [1, 0], No)
Check positivity before either remainder; no unconditional cancellation of common factors is permitted. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, 16, 27, Yes, [1, 0], No).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-remainder-gate · candidate
5. Positive-Modulus Guard before Natural Remainder Comparison
Fix modulus n=4, natural a=19, b=32. Give (n, a, b, n>0?, [a mod n,b mod n] or not-evaluated, congruent? or not-evaluated). This entry excludes zero-modulus conventions.
View answer and explanation
(4, 19, 32, Yes, [3, 0], No)
Check positivity before either remainder; no unconditional cancellation of common factors is permitted. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 19, 32, Yes, [3, 0], No).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-remainder-gate · candidate
6. Positive-Modulus Guard before Natural Remainder Comparison
Fix modulus n=5, natural a=22, b=37. Give (n, a, b, n>0?, [a mod n,b mod n] or not-evaluated, congruent? or not-evaluated). This entry excludes zero-modulus conventions.
View answer and explanation
(5, 22, 37, Yes, [2, 2], Yes)
Check positivity before either remainder; no unconditional cancellation of common factors is permitted. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, 22, 37, Yes, [2, 2], Yes).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-remainder-gate · candidate
7. Positive-Modulus Guard before Natural Remainder Comparison
Fix modulus n=0, natural a=25, b=42. Give (n, a, b, n>0?, [a mod n,b mod n] or not-evaluated, congruent? or not-evaluated). This entry excludes zero-modulus conventions.
View answer and explanation
(0, 25, 42, No, not-evaluated, not-evaluated)
Check positivity before either remainder; no unconditional cancellation of common factors is permitted. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (0, 25, 42, No, not-evaluated, not-evaluated).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-remainder-gate · candidate
8. Positive-Modulus Guard before Natural Remainder Comparison
Fix modulus n=1, natural a=28, b=47. Give (n, a, b, n>0?, [a mod n,b mod n] or not-evaluated, congruent? or not-evaluated). This entry excludes zero-modulus conventions.
View answer and explanation
(1, 28, 47, Yes, [0, 0], Yes)
Check positivity before either remainder; no unconditional cancellation of common factors is permitted. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (1, 28, 47, Yes, [0, 0], Yes).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-remainder-gate · candidate
9. Positive-Modulus Guard before Natural Remainder Comparison
Fix modulus n=2, natural a=31, b=52. Give (n, a, b, n>0?, [a mod n,b mod n] or not-evaluated, congruent? or not-evaluated). This entry excludes zero-modulus conventions.
View answer and explanation
(2, 31, 52, Yes, [1, 0], No)
Check positivity before either remainder; no unconditional cancellation of common factors is permitted. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (2, 31, 52, Yes, [1, 0], No).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-remainder-gate · candidate
10. Positive-Modulus Guard before Natural Remainder Comparison
Fix modulus n=3, natural a=34, b=57. Give (n, a, b, n>0?, [a mod n,b mod n] or not-evaluated, congruent? or not-evaluated). This entry excludes zero-modulus conventions.
View answer and explanation
(3, 34, 57, Yes, [1, 0], No)
Check positivity before either remainder; no unconditional cancellation of common factors is permitted. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, 34, 57, Yes, [1, 0], No).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-remainder-gate · candidate
11. Positive-Modulus Guard before Natural Remainder Comparison
Fix modulus n=4, natural a=37, b=62. Give (n, a, b, n>0?, [a mod n,b mod n] or not-evaluated, congruent? or not-evaluated). This entry excludes zero-modulus conventions.
View answer and explanation
(4, 37, 62, Yes, [1, 2], No)
Check positivity before either remainder; no unconditional cancellation of common factors is permitted. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 37, 62, Yes, [1, 2], No).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-remainder-gate · candidate
12. Positive-Modulus Guard before Natural Remainder Comparison
Fix modulus n=5, natural a=40, b=67. Give (n, a, b, n>0?, [a mod n,b mod n] or not-evaluated, congruent? or not-evaluated). This entry excludes zero-modulus conventions.
View answer and explanation
(5, 40, 67, Yes, [0, 2], No)
Check positivity before either remainder; no unconditional cancellation of common factors is permitted. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, 40, 67, Yes, [0, 2], No).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-remainder-gate · candidate
13. Positive-Modulus Guard before Natural Remainder Comparison
Fix modulus n=0, natural a=43, b=72. Give (n, a, b, n>0?, [a mod n,b mod n] or not-evaluated, congruent? or not-evaluated). This entry excludes zero-modulus conventions.
View answer and explanation
(0, 43, 72, No, not-evaluated, not-evaluated)
Check positivity before either remainder; no unconditional cancellation of common factors is permitted. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (0, 43, 72, No, not-evaluated, not-evaluated).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-remainder-gate · candidate
14. Positive-Modulus Guard before Natural Remainder Comparison
Fix modulus n=1, natural a=46, b=77. Give (n, a, b, n>0?, [a mod n,b mod n] or not-evaluated, congruent? or not-evaluated). This entry excludes zero-modulus conventions.
View answer and explanation
(1, 46, 77, Yes, [0, 0], Yes)
Check positivity before either remainder; no unconditional cancellation of common factors is permitted. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (1, 46, 77, Yes, [0, 0], Yes).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-remainder-gate · candidate
15. Positive-Modulus Guard before Natural Remainder Comparison
Fix modulus n=2, natural a=49, b=82. Give (n, a, b, n>0?, [a mod n,b mod n] or not-evaluated, congruent? or not-evaluated). This entry excludes zero-modulus conventions.
View answer and explanation
(2, 49, 82, Yes, [1, 0], No)
Check positivity before either remainder; no unconditional cancellation of common factors is permitted. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (2, 49, 82, Yes, [1, 0], No).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-remainder-gate · candidate
16. Positive-Modulus Guard before Natural Remainder Comparison
Fix modulus n=3, natural a=52, b=87. Give (n, a, b, n>0?, [a mod n,b mod n] or not-evaluated, congruent? or not-evaluated). This entry excludes zero-modulus conventions.
View answer and explanation
(3, 52, 87, Yes, [1, 0], No)
Check positivity before either remainder; no unconditional cancellation of common factors is permitted. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, 52, 87, Yes, [1, 0], No).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-remainder-gate · candidate
17. Positive-Modulus Guard before Natural Remainder Comparison
Fix modulus n=4, natural a=55, b=92. Give (n, a, b, n>0?, [a mod n,b mod n] or not-evaluated, congruent? or not-evaluated). This entry excludes zero-modulus conventions.
View answer and explanation
(4, 55, 92, Yes, [3, 0], No)
Check positivity before either remainder; no unconditional cancellation of common factors is permitted. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 55, 92, Yes, [3, 0], No).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-remainder-gate · candidate
18. Positive-Modulus Guard before Natural Remainder Comparison
Fix modulus n=5, natural a=58, b=97. Give (n, a, b, n>0?, [a mod n,b mod n] or not-evaluated, congruent? or not-evaluated). This entry excludes zero-modulus conventions.
View answer and explanation
(5, 58, 97, Yes, [3, 2], No)
Check positivity before either remainder; no unconditional cancellation of common factors is permitted. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, 58, 97, Yes, [3, 2], No).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-remainder-gate · candidate
19. Positive-Modulus Guard before Natural Remainder Comparison
Fix modulus n=0, natural a=61, b=102. Give (n, a, b, n>0?, [a mod n,b mod n] or not-evaluated, congruent? or not-evaluated). This entry excludes zero-modulus conventions.
View answer and explanation
(0, 61, 102, No, not-evaluated, not-evaluated)
Check positivity before either remainder; no unconditional cancellation of common factors is permitted. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (0, 61, 102, No, not-evaluated, not-evaluated).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-remainder-gate · candidate
20. Positive-Modulus Guard before Natural Remainder Comparison
Fix modulus n=1, natural a=64, b=107. Give (n, a, b, n>0?, [a mod n,b mod n] or not-evaluated, congruent? or not-evaluated). This entry excludes zero-modulus conventions.
View answer and explanation
(1, 64, 107, Yes, [0, 0], Yes)
Check positivity before either remainder; no unconditional cancellation of common factors is permitted. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (1, 64, 107, Yes, [0, 0], Yes).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-remainder-gate · candidate
21. Positive-Modulus Guard before Natural Remainder Comparison
Fix modulus n=2, natural a=67, b=112. Give (n, a, b, n>0?, [a mod n,b mod n] or not-evaluated, congruent? or not-evaluated). This entry excludes zero-modulus conventions.
View answer and explanation
(2, 67, 112, Yes, [1, 0], No)
Check positivity before either remainder; no unconditional cancellation of common factors is permitted. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (2, 67, 112, Yes, [1, 0], No).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-remainder-gate · candidate
22. Positive-Modulus Guard before Natural Remainder Comparison
Fix modulus n=3, natural a=70, b=117. Give (n, a, b, n>0?, [a mod n,b mod n] or not-evaluated, congruent? or not-evaluated). This entry excludes zero-modulus conventions.
View answer and explanation
(3, 70, 117, Yes, [1, 0], No)
Check positivity before either remainder; no unconditional cancellation of common factors is permitted. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, 70, 117, Yes, [1, 0], No).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-remainder-gate · candidate
23. Positive-Modulus Guard before Natural Remainder Comparison
Fix modulus n=4, natural a=73, b=122. Give (n, a, b, n>0?, [a mod n,b mod n] or not-evaluated, congruent? or not-evaluated). This entry excludes zero-modulus conventions.
View answer and explanation
(4, 73, 122, Yes, [1, 2], No)
Check positivity before either remainder; no unconditional cancellation of common factors is permitted. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 73, 122, Yes, [1, 2], No).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-remainder-gate · candidate
24. Positive-Modulus Guard before Natural Remainder Comparison
Fix modulus n=5, natural a=76, b=127. Give (n, a, b, n>0?, [a mod n,b mod n] or not-evaluated, congruent? or not-evaluated). This entry excludes zero-modulus conventions.
View answer and explanation
(5, 76, 127, Yes, [1, 2], No)
Check positivity before either remainder; no unconditional cancellation of common factors is permitted. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, 76, 127, Yes, [1, 2], No).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-remainder-gate · candidate
25. Positive-Modulus Guard before Natural Remainder Comparison
Fix modulus n=0, natural a=79, b=132. Give (n, a, b, n>0?, [a mod n,b mod n] or not-evaluated, congruent? or not-evaluated). This entry excludes zero-modulus conventions.
View answer and explanation
(0, 79, 132, No, not-evaluated, not-evaluated)
Check positivity before either remainder; no unconditional cancellation of common factors is permitted. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (0, 79, 132, No, not-evaluated, not-evaluated).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-remainder-gate · candidate
26. A Natural Congruence Class in an Inclusive Window
Fix positive modulus n=2, natural a=1, bound=25. Give (n, a, bound, a mod n, every x∈[0,bound] with x mod n=a mod n).
View answer and explanation
(2, 1, 25, 1, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25])
Use the same positive modulus for the entire window; congruence is not numerical equality. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (2, 1, 25, 1, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25]).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-equivalence-class · candidate
27. A Natural Congruence Class in an Inclusive Window
Fix positive modulus n=3, natural a=4, bound=26. Give (n, a, bound, a mod n, every x∈[0,bound] with x mod n=a mod n).
View answer and explanation
(3, 4, 26, 1, [1, 4, 7, 10, 13, 16, 19, 22, 25])
Use the same positive modulus for the entire window; congruence is not numerical equality. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, 4, 26, 1, [1, 4, 7, 10, 13, 16, 19, 22, 25]).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-equivalence-class · candidate
28. A Natural Congruence Class in an Inclusive Window
Fix positive modulus n=4, natural a=7, bound=27. Give (n, a, bound, a mod n, every x∈[0,bound] with x mod n=a mod n).
View answer and explanation
(4, 7, 27, 3, [3, 7, 11, 15, 19, 23, 27])
Use the same positive modulus for the entire window; congruence is not numerical equality. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 7, 27, 3, [3, 7, 11, 15, 19, 23, 27]).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-equivalence-class · candidate
29. A Natural Congruence Class in an Inclusive Window
Fix positive modulus n=5, natural a=10, bound=28. Give (n, a, bound, a mod n, every x∈[0,bound] with x mod n=a mod n).
View answer and explanation
(5, 10, 28, 0, [0, 5, 10, 15, 20, 25])
Use the same positive modulus for the entire window; congruence is not numerical equality. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, 10, 28, 0, [0, 5, 10, 15, 20, 25]).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-equivalence-class · candidate
30. A Natural Congruence Class in an Inclusive Window
Fix positive modulus n=6, natural a=13, bound=29. Give (n, a, bound, a mod n, every x∈[0,bound] with x mod n=a mod n).
View answer and explanation
(6, 13, 29, 1, [1, 7, 13, 19, 25])
Use the same positive modulus for the entire window; congruence is not numerical equality. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, 13, 29, 1, [1, 7, 13, 19, 25]).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-equivalence-class · candidate
31. A Natural Congruence Class in an Inclusive Window
Fix positive modulus n=7, natural a=16, bound=30. Give (n, a, bound, a mod n, every x∈[0,bound] with x mod n=a mod n).
View answer and explanation
(7, 16, 30, 2, [2, 9, 16, 23, 30])
Use the same positive modulus for the entire window; congruence is not numerical equality. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (7, 16, 30, 2, [2, 9, 16, 23, 30]).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-equivalence-class · candidate
32. A Natural Congruence Class in an Inclusive Window
Fix positive modulus n=8, natural a=19, bound=31. Give (n, a, bound, a mod n, every x∈[0,bound] with x mod n=a mod n).
View answer and explanation
(8, 19, 31, 3, [3, 11, 19, 27])
Use the same positive modulus for the entire window; congruence is not numerical equality. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (8, 19, 31, 3, [3, 11, 19, 27]).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-equivalence-class · candidate
33. A Natural Congruence Class in an Inclusive Window
Fix positive modulus n=9, natural a=22, bound=32. Give (n, a, bound, a mod n, every x∈[0,bound] with x mod n=a mod n).
View answer and explanation
(9, 22, 32, 4, [4, 13, 22, 31])
Use the same positive modulus for the entire window; congruence is not numerical equality. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (9, 22, 32, 4, [4, 13, 22, 31]).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-equivalence-class · candidate
34. A Natural Congruence Class in an Inclusive Window
Fix positive modulus n=10, natural a=25, bound=33. Give (n, a, bound, a mod n, every x∈[0,bound] with x mod n=a mod n).
View answer and explanation
(10, 25, 33, 5, [5, 15, 25])
Use the same positive modulus for the entire window; congruence is not numerical equality. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (10, 25, 33, 5, [5, 15, 25]).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-equivalence-class · candidate
35. A Natural Congruence Class in an Inclusive Window
Fix positive modulus n=2, natural a=28, bound=34. Give (n, a, bound, a mod n, every x∈[0,bound] with x mod n=a mod n).
View answer and explanation
(2, 28, 34, 0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34])
Use the same positive modulus for the entire window; congruence is not numerical equality. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (2, 28, 34, 0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34]).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-equivalence-class · candidate
36. A Natural Congruence Class in an Inclusive Window
Fix positive modulus n=3, natural a=31, bound=35. Give (n, a, bound, a mod n, every x∈[0,bound] with x mod n=a mod n).
View answer and explanation
(3, 31, 35, 1, [1, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34])
Use the same positive modulus for the entire window; congruence is not numerical equality. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, 31, 35, 1, [1, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34]).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-equivalence-class · candidate
37. A Natural Congruence Class in an Inclusive Window
Fix positive modulus n=4, natural a=34, bound=36. Give (n, a, bound, a mod n, every x∈[0,bound] with x mod n=a mod n).
View answer and explanation
(4, 34, 36, 2, [2, 6, 10, 14, 18, 22, 26, 30, 34])
Use the same positive modulus for the entire window; congruence is not numerical equality. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 34, 36, 2, [2, 6, 10, 14, 18, 22, 26, 30, 34]).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-equivalence-class · candidate
38. A Natural Congruence Class in an Inclusive Window
Fix positive modulus n=5, natural a=37, bound=37. Give (n, a, bound, a mod n, every x∈[0,bound] with x mod n=a mod n).
View answer and explanation
(5, 37, 37, 2, [2, 7, 12, 17, 22, 27, 32, 37])
Use the same positive modulus for the entire window; congruence is not numerical equality. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, 37, 37, 2, [2, 7, 12, 17, 22, 27, 32, 37]).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-equivalence-class · candidate
39. A Natural Congruence Class in an Inclusive Window
Fix positive modulus n=6, natural a=40, bound=38. Give (n, a, bound, a mod n, every x∈[0,bound] with x mod n=a mod n).
View answer and explanation
(6, 40, 38, 4, [4, 10, 16, 22, 28, 34])
Use the same positive modulus for the entire window; congruence is not numerical equality. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, 40, 38, 4, [4, 10, 16, 22, 28, 34]).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-equivalence-class · candidate
40. A Natural Congruence Class in an Inclusive Window
Fix positive modulus n=7, natural a=43, bound=39. Give (n, a, bound, a mod n, every x∈[0,bound] with x mod n=a mod n).
View answer and explanation
(7, 43, 39, 1, [1, 8, 15, 22, 29, 36])
Use the same positive modulus for the entire window; congruence is not numerical equality. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (7, 43, 39, 1, [1, 8, 15, 22, 29, 36]).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-equivalence-class · candidate
41. A Natural Congruence Class in an Inclusive Window
Fix positive modulus n=8, natural a=46, bound=40. Give (n, a, bound, a mod n, every x∈[0,bound] with x mod n=a mod n).
View answer and explanation
(8, 46, 40, 6, [6, 14, 22, 30, 38])
Use the same positive modulus for the entire window; congruence is not numerical equality. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (8, 46, 40, 6, [6, 14, 22, 30, 38]).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-equivalence-class · candidate
42. A Natural Congruence Class in an Inclusive Window
Fix positive modulus n=9, natural a=49, bound=41. Give (n, a, bound, a mod n, every x∈[0,bound] with x mod n=a mod n).
View answer and explanation
(9, 49, 41, 4, [4, 13, 22, 31, 40])
Use the same positive modulus for the entire window; congruence is not numerical equality. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (9, 49, 41, 4, [4, 13, 22, 31, 40]).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-equivalence-class · candidate
43. A Natural Congruence Class in an Inclusive Window
Fix positive modulus n=10, natural a=52, bound=42. Give (n, a, bound, a mod n, every x∈[0,bound] with x mod n=a mod n).
View answer and explanation
(10, 52, 42, 2, [2, 12, 22, 32, 42])
Use the same positive modulus for the entire window; congruence is not numerical equality. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (10, 52, 42, 2, [2, 12, 22, 32, 42]).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-equivalence-class · candidate
44. A Natural Congruence Class in an Inclusive Window
Fix positive modulus n=2, natural a=55, bound=43. Give (n, a, bound, a mod n, every x∈[0,bound] with x mod n=a mod n).
View answer and explanation
(2, 55, 43, 1, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43])
Use the same positive modulus for the entire window; congruence is not numerical equality. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (2, 55, 43, 1, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43]).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-equivalence-class · candidate
45. A Natural Congruence Class in an Inclusive Window
Fix positive modulus n=3, natural a=58, bound=44. Give (n, a, bound, a mod n, every x∈[0,bound] with x mod n=a mod n).
View answer and explanation
(3, 58, 44, 1, [1, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34, 37, 40, 43])
Use the same positive modulus for the entire window; congruence is not numerical equality. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, 58, 44, 1, [1, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34, 37, 40, 43]).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-equivalence-class · candidate
46. A Natural Congruence Class in an Inclusive Window
Fix positive modulus n=4, natural a=61, bound=45. Give (n, a, bound, a mod n, every x∈[0,bound] with x mod n=a mod n).
View answer and explanation
(4, 61, 45, 1, [1, 5, 9, 13, 17, 21, 25, 29, 33, 37, 41, 45])
Use the same positive modulus for the entire window; congruence is not numerical equality. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 61, 45, 1, [1, 5, 9, 13, 17, 21, 25, 29, 33, 37, 41, 45]).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-equivalence-class · candidate
47. A Natural Congruence Class in an Inclusive Window
Fix positive modulus n=5, natural a=64, bound=46. Give (n, a, bound, a mod n, every x∈[0,bound] with x mod n=a mod n).
View answer and explanation
(5, 64, 46, 4, [4, 9, 14, 19, 24, 29, 34, 39, 44])
Use the same positive modulus for the entire window; congruence is not numerical equality. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, 64, 46, 4, [4, 9, 14, 19, 24, 29, 34, 39, 44]).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-equivalence-class · candidate
48. A Natural Congruence Class in an Inclusive Window
Fix positive modulus n=6, natural a=67, bound=47. Give (n, a, bound, a mod n, every x∈[0,bound] with x mod n=a mod n).
View answer and explanation
(6, 67, 47, 1, [1, 7, 13, 19, 25, 31, 37, 43])
Use the same positive modulus for the entire window; congruence is not numerical equality. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, 67, 47, 1, [1, 7, 13, 19, 25, 31, 37, 43]).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-equivalence-class · candidate
49. A Natural Congruence Class in an Inclusive Window
Fix positive modulus n=7, natural a=70, bound=48. Give (n, a, bound, a mod n, every x∈[0,bound] with x mod n=a mod n).
View answer and explanation
(7, 70, 48, 0, [0, 7, 14, 21, 28, 35, 42])
Use the same positive modulus for the entire window; congruence is not numerical equality. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (7, 70, 48, 0, [0, 7, 14, 21, 28, 35, 42]).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-equivalence-class · candidate
50. A Natural Congruence Class in an Inclusive Window
Fix positive modulus n=8, natural a=73, bound=49. Give (n, a, bound, a mod n, every x∈[0,bound] with x mod n=a mod n).
View answer and explanation
(8, 73, 49, 1, [1, 9, 17, 25, 33, 41, 49])
Use the same positive modulus for the entire window; congruence is not numerical equality. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (8, 73, 49, 1, [1, 9, 17, 25, 33, 41, 49]).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-equivalence-class · candidate
51. Additive Congruence Values, Premises, and Material Implication
Use fixed positive modulus n=2, natural inputs a=1, b=3, c=3, d=7. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a+c) mod n,(b+d) mod n], sum congruence?, premises imply conclusion?).
View answer and explanation
(2, [1, 3], [3, 7], Yes, [0, 0], Yes, Yes)
Recompute both premises and the conclusion separately. A false premise can make the material implication true even when it supplies no applicable preservation premise. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (2, [1, 3], [3, 7], Yes, [0, 0], Yes, Yes).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-add-preservation · candidate
52. Additive Congruence Values, Premises, and Material Implication
Use fixed positive modulus n=3, natural inputs a=2, b=8, c=5, d=14. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a+c) mod n,(b+d) mod n], sum congruence?, premises imply conclusion?).
View answer and explanation
(3, [2, 8], [5, 14], Yes, [1, 1], Yes, Yes)
Recompute both premises and the conclusion separately. A false premise can make the material implication true even when it supplies no applicable preservation premise. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, [2, 8], [5, 14], Yes, [1, 1], Yes, Yes).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-add-preservation · candidate
53. Additive Congruence Values, Premises, and Material Implication
Use fixed positive modulus n=4, natural inputs a=3, b=15, c=7, d=11. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a+c) mod n,(b+d) mod n], sum congruence?, premises imply conclusion?).
View answer and explanation
(4, [3, 15], [7, 11], Yes, [2, 2], Yes, Yes)
Recompute both premises and the conclusion separately. A false premise can make the material implication true even when it supplies no applicable preservation premise. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [3, 15], [7, 11], Yes, [2, 2], Yes, Yes).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-add-preservation · candidate
54. Additive Congruence Values, Premises, and Material Implication
Use fixed positive modulus n=5, natural inputs a=4, b=9, c=9, d=19. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a+c) mod n,(b+d) mod n], sum congruence?, premises imply conclusion?).
View answer and explanation
(5, [4, 9], [9, 19], Yes, [3, 3], Yes, Yes)
Recompute both premises and the conclusion separately. A false premise can make the material implication true even when it supplies no applicable preservation premise. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [4, 9], [9, 19], Yes, [3, 3], Yes, Yes).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-add-preservation · candidate
55. Additive Congruence Values, Premises, and Material Implication
Use fixed positive modulus n=6, natural inputs a=5, b=17, c=11, d=29. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a+c) mod n,(b+d) mod n], sum congruence?, premises imply conclusion?).
View answer and explanation
(6, [5, 17], [11, 29], Yes, [4, 4], Yes, Yes)
Recompute both premises and the conclusion separately. A false premise can make the material implication true even when it supplies no applicable preservation premise. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [5, 17], [11, 29], Yes, [4, 4], Yes, Yes).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-add-preservation · candidate
56. Additive Congruence Values, Premises, and Material Implication
Use fixed positive modulus n=7, natural inputs a=6, b=27, c=13, d=20. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a+c) mod n,(b+d) mod n], sum congruence?, premises imply conclusion?).
View answer and explanation
(7, [6, 27], [13, 20], Yes, [5, 5], Yes, Yes)
Recompute both premises and the conclusion separately. A false premise can make the material implication true even when it supplies no applicable preservation premise. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (7, [6, 27], [13, 20], Yes, [5, 5], Yes, Yes).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-add-preservation · candidate
57. Additive Congruence Values, Premises, and Material Implication
Use fixed positive modulus n=8, natural inputs a=7, b=15, c=15, d=31. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a+c) mod n,(b+d) mod n], sum congruence?, premises imply conclusion?).
View answer and explanation
(8, [7, 15], [15, 31], Yes, [6, 6], Yes, Yes)
Recompute both premises and the conclusion separately. A false premise can make the material implication true even when it supplies no applicable preservation premise. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (8, [7, 15], [15, 31], Yes, [6, 6], Yes, Yes).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-add-preservation · candidate
58. Additive Congruence Values, Premises, and Material Implication
Use fixed positive modulus n=9, natural inputs a=8, b=26, c=17, d=44. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a+c) mod n,(b+d) mod n], sum congruence?, premises imply conclusion?).
View answer and explanation
(9, [8, 26], [17, 44], Yes, [7, 7], Yes, Yes)
Recompute both premises and the conclusion separately. A false premise can make the material implication true even when it supplies no applicable preservation premise. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (9, [8, 26], [17, 44], Yes, [7, 7], Yes, Yes).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-add-preservation · candidate
59. Additive Congruence Values, Premises, and Material Implication
Use fixed positive modulus n=10, natural inputs a=9, b=39, c=19, d=29. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a+c) mod n,(b+d) mod n], sum congruence?, premises imply conclusion?).
View answer and explanation
(10, [9, 39], [19, 29], Yes, [8, 8], Yes, Yes)
Recompute both premises and the conclusion separately. A false premise can make the material implication true even when it supplies no applicable preservation premise. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (10, [9, 39], [19, 29], Yes, [8, 8], Yes, Yes).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-add-preservation · candidate
60. Additive Congruence Values, Premises, and Material Implication
Use fixed positive modulus n=2, natural inputs a=10, b=12, c=21, d=25. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a+c) mod n,(b+d) mod n], sum congruence?, premises imply conclusion?).
View answer and explanation
(2, [10, 12], [21, 25], Yes, [1, 1], Yes, Yes)
Recompute both premises and the conclusion separately. A false premise can make the material implication true even when it supplies no applicable preservation premise. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (2, [10, 12], [21, 25], Yes, [1, 1], Yes, Yes).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-add-preservation · candidate
61. Additive Congruence Values, Premises, and Material Implication
Use fixed positive modulus n=3, natural inputs a=11, b=17, c=23, d=32. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a+c) mod n,(b+d) mod n], sum congruence?, premises imply conclusion?).
View answer and explanation
(3, [11, 17], [23, 32], Yes, [1, 1], Yes, Yes)
Recompute both premises and the conclusion separately. A false premise can make the material implication true even when it supplies no applicable preservation premise. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, [11, 17], [23, 32], Yes, [1, 1], Yes, Yes).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-add-preservation · candidate
62. Additive Congruence Values, Premises, and Material Implication
Use fixed positive modulus n=4, natural inputs a=12, b=24, c=25, d=29. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a+c) mod n,(b+d) mod n], sum congruence?, premises imply conclusion?).
View answer and explanation
(4, [12, 24], [25, 29], Yes, [1, 1], Yes, Yes)
Recompute both premises and the conclusion separately. A false premise can make the material implication true even when it supplies no applicable preservation premise. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [12, 24], [25, 29], Yes, [1, 1], Yes, Yes).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-add-preservation · candidate
63. Additive Congruence Values, Premises, and Material Implication
Use fixed positive modulus n=5, natural inputs a=13, b=18, c=27, d=37. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a+c) mod n,(b+d) mod n], sum congruence?, premises imply conclusion?).
View answer and explanation
(5, [13, 18], [27, 37], Yes, [0, 0], Yes, Yes)
Recompute both premises and the conclusion separately. A false premise can make the material implication true even when it supplies no applicable preservation premise. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [13, 18], [27, 37], Yes, [0, 0], Yes, Yes).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-add-preservation · candidate
64. Additive Congruence Values, Premises, and Material Implication
Use fixed positive modulus n=6, natural inputs a=14, b=26, c=29, d=47. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a+c) mod n,(b+d) mod n], sum congruence?, premises imply conclusion?).
View answer and explanation
(6, [14, 26], [29, 47], Yes, [1, 1], Yes, Yes)
Recompute both premises and the conclusion separately. A false premise can make the material implication true even when it supplies no applicable preservation premise. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [14, 26], [29, 47], Yes, [1, 1], Yes, Yes).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-add-preservation · candidate
65. Additive Congruence Values, Premises, and Material Implication
Use fixed positive modulus n=7, natural inputs a=15, b=36, c=31, d=38. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a+c) mod n,(b+d) mod n], sum congruence?, premises imply conclusion?).
View answer and explanation
(7, [15, 36], [31, 38], Yes, [4, 4], Yes, Yes)
Recompute both premises and the conclusion separately. A false premise can make the material implication true even when it supplies no applicable preservation premise. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (7, [15, 36], [31, 38], Yes, [4, 4], Yes, Yes).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-add-preservation · candidate
66. Additive Congruence Values, Premises, and Material Implication
Use fixed positive modulus n=8, natural inputs a=16, b=24, c=33, d=49. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a+c) mod n,(b+d) mod n], sum congruence?, premises imply conclusion?).
View answer and explanation
(8, [16, 24], [33, 49], Yes, [1, 1], Yes, Yes)
Recompute both premises and the conclusion separately. A false premise can make the material implication true even when it supplies no applicable preservation premise. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (8, [16, 24], [33, 49], Yes, [1, 1], Yes, Yes).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-add-preservation · candidate
67. Additive Congruence Values, Premises, and Material Implication
Use fixed positive modulus n=9, natural inputs a=17, b=35, c=35, d=62. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a+c) mod n,(b+d) mod n], sum congruence?, premises imply conclusion?).
View answer and explanation
(9, [17, 35], [35, 62], Yes, [7, 7], Yes, Yes)
Recompute both premises and the conclusion separately. A false premise can make the material implication true even when it supplies no applicable preservation premise. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (9, [17, 35], [35, 62], Yes, [7, 7], Yes, Yes).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-add-preservation · candidate
68. Additive Congruence Values, Premises, and Material Implication
Use fixed positive modulus n=10, natural inputs a=18, b=48, c=37, d=47. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a+c) mod n,(b+d) mod n], sum congruence?, premises imply conclusion?).
View answer and explanation
(10, [18, 48], [37, 47], Yes, [5, 5], Yes, Yes)
Recompute both premises and the conclusion separately. A false premise can make the material implication true even when it supplies no applicable preservation premise. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (10, [18, 48], [37, 47], Yes, [5, 5], Yes, Yes).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-add-preservation · candidate
69. Additive Congruence Values, Premises, and Material Implication
Use fixed positive modulus n=2, natural inputs a=19, b=21, c=39, d=43. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a+c) mod n,(b+d) mod n], sum congruence?, premises imply conclusion?).
View answer and explanation
(2, [19, 21], [39, 43], Yes, [0, 0], Yes, Yes)
Recompute both premises and the conclusion separately. A false premise can make the material implication true even when it supplies no applicable preservation premise. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (2, [19, 21], [39, 43], Yes, [0, 0], Yes, Yes).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-add-preservation · candidate
70. Additive Congruence Values, Premises, and Material Implication
Use fixed positive modulus n=3, natural inputs a=20, b=26, c=41, d=50. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a+c) mod n,(b+d) mod n], sum congruence?, premises imply conclusion?).
View answer and explanation
(3, [20, 26], [41, 50], Yes, [1, 1], Yes, Yes)
Recompute both premises and the conclusion separately. A false premise can make the material implication true even when it supplies no applicable preservation premise. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, [20, 26], [41, 50], Yes, [1, 1], Yes, Yes).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-add-preservation · candidate
71. Additive Congruence Values, Premises, and Material Implication
Use fixed positive modulus n=4, natural inputs a=21, b=33, c=43, d=47. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a+c) mod n,(b+d) mod n], sum congruence?, premises imply conclusion?).
View answer and explanation
(4, [21, 33], [43, 47], Yes, [0, 0], Yes, Yes)
Recompute both premises and the conclusion separately. A false premise can make the material implication true even when it supplies no applicable preservation premise. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [21, 33], [43, 47], Yes, [0, 0], Yes, Yes).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-add-preservation · candidate
72. Additive Congruence Values, Premises, and Material Implication
Use fixed positive modulus n=5, natural inputs a=22, b=27, c=45, d=55. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a+c) mod n,(b+d) mod n], sum congruence?, premises imply conclusion?).
View answer and explanation
(5, [22, 27], [45, 55], Yes, [2, 2], Yes, Yes)
Recompute both premises and the conclusion separately. A false premise can make the material implication true even when it supplies no applicable preservation premise. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [22, 27], [45, 55], Yes, [2, 2], Yes, Yes).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-add-preservation · candidate
73. Additive Congruence Values, Premises, and Material Implication
Use fixed positive modulus n=6, natural inputs a=23, b=35, c=47, d=65. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a+c) mod n,(b+d) mod n], sum congruence?, premises imply conclusion?).
View answer and explanation
(6, [23, 35], [47, 65], Yes, [4, 4], Yes, Yes)
Recompute both premises and the conclusion separately. A false premise can make the material implication true even when it supplies no applicable preservation premise. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [23, 35], [47, 65], Yes, [4, 4], Yes, Yes).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-add-preservation · candidate
74. Additive Congruence Values, Premises, and Material Implication
Use fixed positive modulus n=7, natural inputs a=24, b=45, c=49, d=56. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a+c) mod n,(b+d) mod n], sum congruence?, premises imply conclusion?).
View answer and explanation
(7, [24, 45], [49, 56], Yes, [3, 3], Yes, Yes)
Recompute both premises and the conclusion separately. A false premise can make the material implication true even when it supplies no applicable preservation premise. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (7, [24, 45], [49, 56], Yes, [3, 3], Yes, Yes).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-add-preservation · candidate
75. Additive Congruence Values, Premises, and Material Implication
Use fixed positive modulus n=8, natural inputs a=25, b=33, c=51, d=67. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a+c) mod n,(b+d) mod n], sum congruence?, premises imply conclusion?).
View answer and explanation
(8, [25, 33], [51, 67], Yes, [4, 4], Yes, Yes)
Recompute both premises and the conclusion separately. A false premise can make the material implication true even when it supplies no applicable preservation premise. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (8, [25, 33], [51, 67], Yes, [4, 4], Yes, Yes).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-add-preservation · candidate
76. Multiplicative Congruence Values, Premises, and Material Implication
Use fixed positive modulus n=2, natural inputs a=1, b=3, c=3, d=7. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a*c) mod n,(b*d) mod n], product congruence?, premises imply conclusion?). Do not cancel factors without additional conditions.
View answer and explanation
(2, [1, 3], [3, 7], Yes, [1, 1], Yes, Yes)
Calculate both product remainders and keep premise truth, conclusion truth and material implication distinct. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (2, [1, 3], [3, 7], Yes, [1, 1], Yes, Yes).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-mul-preservation · candidate
77. Multiplicative Congruence Values, Premises, and Material Implication
Use fixed positive modulus n=3, natural inputs a=2, b=8, c=5, d=14. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a*c) mod n,(b*d) mod n], product congruence?, premises imply conclusion?). Do not cancel factors without additional conditions.
View answer and explanation
(3, [2, 8], [5, 14], Yes, [1, 1], Yes, Yes)
Calculate both product remainders and keep premise truth, conclusion truth and material implication distinct. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, [2, 8], [5, 14], Yes, [1, 1], Yes, Yes).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-mul-preservation · candidate
78. Multiplicative Congruence Values, Premises, and Material Implication
Use fixed positive modulus n=4, natural inputs a=3, b=15, c=7, d=11. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a*c) mod n,(b*d) mod n], product congruence?, premises imply conclusion?). Do not cancel factors without additional conditions.
View answer and explanation
(4, [3, 15], [7, 11], Yes, [1, 1], Yes, Yes)
Calculate both product remainders and keep premise truth, conclusion truth and material implication distinct. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [3, 15], [7, 11], Yes, [1, 1], Yes, Yes).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-mul-preservation · candidate
79. Multiplicative Congruence Values, Premises, and Material Implication
Use fixed positive modulus n=5, natural inputs a=4, b=9, c=9, d=19. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a*c) mod n,(b*d) mod n], product congruence?, premises imply conclusion?). Do not cancel factors without additional conditions.
View answer and explanation
(5, [4, 9], [9, 19], Yes, [1, 1], Yes, Yes)
Calculate both product remainders and keep premise truth, conclusion truth and material implication distinct. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [4, 9], [9, 19], Yes, [1, 1], Yes, Yes).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-mul-preservation · candidate
80. Multiplicative Congruence Values, Premises, and Material Implication
Use fixed positive modulus n=6, natural inputs a=5, b=17, c=11, d=29. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a*c) mod n,(b*d) mod n], product congruence?, premises imply conclusion?). Do not cancel factors without additional conditions.
View answer and explanation
(6, [5, 17], [11, 29], Yes, [1, 1], Yes, Yes)
Calculate both product remainders and keep premise truth, conclusion truth and material implication distinct. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [5, 17], [11, 29], Yes, [1, 1], Yes, Yes).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-mul-preservation · candidate
81. Multiplicative Congruence Values, Premises, and Material Implication
Use fixed positive modulus n=7, natural inputs a=6, b=27, c=13, d=20. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a*c) mod n,(b*d) mod n], product congruence?, premises imply conclusion?). Do not cancel factors without additional conditions.
View answer and explanation
(7, [6, 27], [13, 20], Yes, [1, 1], Yes, Yes)
Calculate both product remainders and keep premise truth, conclusion truth and material implication distinct. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (7, [6, 27], [13, 20], Yes, [1, 1], Yes, Yes).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-mul-preservation · candidate
82. Multiplicative Congruence Values, Premises, and Material Implication
Use fixed positive modulus n=8, natural inputs a=7, b=15, c=15, d=31. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a*c) mod n,(b*d) mod n], product congruence?, premises imply conclusion?). Do not cancel factors without additional conditions.
View answer and explanation
(8, [7, 15], [15, 31], Yes, [1, 1], Yes, Yes)
Calculate both product remainders and keep premise truth, conclusion truth and material implication distinct. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (8, [7, 15], [15, 31], Yes, [1, 1], Yes, Yes).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-mul-preservation · candidate
83. Multiplicative Congruence Values, Premises, and Material Implication
Use fixed positive modulus n=9, natural inputs a=8, b=26, c=17, d=44. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a*c) mod n,(b*d) mod n], product congruence?, premises imply conclusion?). Do not cancel factors without additional conditions.
View answer and explanation
(9, [8, 26], [17, 44], Yes, [1, 1], Yes, Yes)
Calculate both product remainders and keep premise truth, conclusion truth and material implication distinct. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (9, [8, 26], [17, 44], Yes, [1, 1], Yes, Yes).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-mul-preservation · candidate
84. Multiplicative Congruence Values, Premises, and Material Implication
Use fixed positive modulus n=10, natural inputs a=9, b=39, c=19, d=29. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a*c) mod n,(b*d) mod n], product congruence?, premises imply conclusion?). Do not cancel factors without additional conditions.
View answer and explanation
(10, [9, 39], [19, 29], Yes, [1, 1], Yes, Yes)
Calculate both product remainders and keep premise truth, conclusion truth and material implication distinct. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (10, [9, 39], [19, 29], Yes, [1, 1], Yes, Yes).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-mul-preservation · candidate
85. Multiplicative Congruence Values, Premises, and Material Implication
Use fixed positive modulus n=2, natural inputs a=10, b=12, c=21, d=25. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a*c) mod n,(b*d) mod n], product congruence?, premises imply conclusion?). Do not cancel factors without additional conditions.
View answer and explanation
(2, [10, 12], [21, 25], Yes, [0, 0], Yes, Yes)
Calculate both product remainders and keep premise truth, conclusion truth and material implication distinct. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (2, [10, 12], [21, 25], Yes, [0, 0], Yes, Yes).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-mul-preservation · candidate
86. Multiplicative Congruence Values, Premises, and Material Implication
Use fixed positive modulus n=3, natural inputs a=11, b=17, c=23, d=32. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a*c) mod n,(b*d) mod n], product congruence?, premises imply conclusion?). Do not cancel factors without additional conditions.
View answer and explanation
(3, [11, 17], [23, 32], Yes, [1, 1], Yes, Yes)
Calculate both product remainders and keep premise truth, conclusion truth and material implication distinct. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, [11, 17], [23, 32], Yes, [1, 1], Yes, Yes).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-mul-preservation · candidate
87. Multiplicative Congruence Values, Premises, and Material Implication
Use fixed positive modulus n=4, natural inputs a=12, b=24, c=25, d=29. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a*c) mod n,(b*d) mod n], product congruence?, premises imply conclusion?). Do not cancel factors without additional conditions.
View answer and explanation
(4, [12, 24], [25, 29], Yes, [0, 0], Yes, Yes)
Calculate both product remainders and keep premise truth, conclusion truth and material implication distinct. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [12, 24], [25, 29], Yes, [0, 0], Yes, Yes).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-mul-preservation · candidate
88. Multiplicative Congruence Values, Premises, and Material Implication
Use fixed positive modulus n=5, natural inputs a=13, b=18, c=27, d=37. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a*c) mod n,(b*d) mod n], product congruence?, premises imply conclusion?). Do not cancel factors without additional conditions.
View answer and explanation
(5, [13, 18], [27, 37], Yes, [1, 1], Yes, Yes)
Calculate both product remainders and keep premise truth, conclusion truth and material implication distinct. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [13, 18], [27, 37], Yes, [1, 1], Yes, Yes).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-mul-preservation · candidate
89. Multiplicative Congruence Values, Premises, and Material Implication
Use fixed positive modulus n=6, natural inputs a=14, b=26, c=29, d=47. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a*c) mod n,(b*d) mod n], product congruence?, premises imply conclusion?). Do not cancel factors without additional conditions.
View answer and explanation
(6, [14, 26], [29, 47], Yes, [4, 4], Yes, Yes)
Calculate both product remainders and keep premise truth, conclusion truth and material implication distinct. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [14, 26], [29, 47], Yes, [4, 4], Yes, Yes).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-mul-preservation · candidate
90. Multiplicative Congruence Values, Premises, and Material Implication
Use fixed positive modulus n=7, natural inputs a=15, b=36, c=31, d=38. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a*c) mod n,(b*d) mod n], product congruence?, premises imply conclusion?). Do not cancel factors without additional conditions.
View answer and explanation
(7, [15, 36], [31, 38], Yes, [3, 3], Yes, Yes)
Calculate both product remainders and keep premise truth, conclusion truth and material implication distinct. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (7, [15, 36], [31, 38], Yes, [3, 3], Yes, Yes).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-mul-preservation · candidate
91. Multiplicative Congruence Values, Premises, and Material Implication
Use fixed positive modulus n=8, natural inputs a=16, b=24, c=33, d=49. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a*c) mod n,(b*d) mod n], product congruence?, premises imply conclusion?). Do not cancel factors without additional conditions.
View answer and explanation
(8, [16, 24], [33, 49], Yes, [0, 0], Yes, Yes)
Calculate both product remainders and keep premise truth, conclusion truth and material implication distinct. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (8, [16, 24], [33, 49], Yes, [0, 0], Yes, Yes).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-mul-preservation · candidate
92. Multiplicative Congruence Values, Premises, and Material Implication
Use fixed positive modulus n=9, natural inputs a=17, b=35, c=35, d=62. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a*c) mod n,(b*d) mod n], product congruence?, premises imply conclusion?). Do not cancel factors without additional conditions.
View answer and explanation
(9, [17, 35], [35, 62], Yes, [1, 1], Yes, Yes)
Calculate both product remainders and keep premise truth, conclusion truth and material implication distinct. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (9, [17, 35], [35, 62], Yes, [1, 1], Yes, Yes).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-mul-preservation · candidate
93. Multiplicative Congruence Values, Premises, and Material Implication
Use fixed positive modulus n=10, natural inputs a=18, b=48, c=37, d=47. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a*c) mod n,(b*d) mod n], product congruence?, premises imply conclusion?). Do not cancel factors without additional conditions.
View answer and explanation
(10, [18, 48], [37, 47], Yes, [6, 6], Yes, Yes)
Calculate both product remainders and keep premise truth, conclusion truth and material implication distinct. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (10, [18, 48], [37, 47], Yes, [6, 6], Yes, Yes).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-mul-preservation · candidate
94. Multiplicative Congruence Values, Premises, and Material Implication
Use fixed positive modulus n=2, natural inputs a=19, b=21, c=39, d=43. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a*c) mod n,(b*d) mod n], product congruence?, premises imply conclusion?). Do not cancel factors without additional conditions.
View answer and explanation
(2, [19, 21], [39, 43], Yes, [1, 1], Yes, Yes)
Calculate both product remainders and keep premise truth, conclusion truth and material implication distinct. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (2, [19, 21], [39, 43], Yes, [1, 1], Yes, Yes).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-mul-preservation · candidate
95. Multiplicative Congruence Values, Premises, and Material Implication
Use fixed positive modulus n=3, natural inputs a=20, b=26, c=41, d=50. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a*c) mod n,(b*d) mod n], product congruence?, premises imply conclusion?). Do not cancel factors without additional conditions.
View answer and explanation
(3, [20, 26], [41, 50], Yes, [1, 1], Yes, Yes)
Calculate both product remainders and keep premise truth, conclusion truth and material implication distinct. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, [20, 26], [41, 50], Yes, [1, 1], Yes, Yes).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-mul-preservation · candidate
96. Multiplicative Congruence Values, Premises, and Material Implication
Use fixed positive modulus n=4, natural inputs a=21, b=33, c=43, d=47. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a*c) mod n,(b*d) mod n], product congruence?, premises imply conclusion?). Do not cancel factors without additional conditions.
View answer and explanation
(4, [21, 33], [43, 47], Yes, [3, 3], Yes, Yes)
Calculate both product remainders and keep premise truth, conclusion truth and material implication distinct. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [21, 33], [43, 47], Yes, [3, 3], Yes, Yes).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-mul-preservation · candidate
97. Multiplicative Congruence Values, Premises, and Material Implication
Use fixed positive modulus n=5, natural inputs a=22, b=27, c=45, d=55. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a*c) mod n,(b*d) mod n], product congruence?, premises imply conclusion?). Do not cancel factors without additional conditions.
View answer and explanation
(5, [22, 27], [45, 55], Yes, [0, 0], Yes, Yes)
Calculate both product remainders and keep premise truth, conclusion truth and material implication distinct. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [22, 27], [45, 55], Yes, [0, 0], Yes, Yes).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-mul-preservation · candidate
98. Multiplicative Congruence Values, Premises, and Material Implication
Use fixed positive modulus n=6, natural inputs a=23, b=35, c=47, d=65. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a*c) mod n,(b*d) mod n], product congruence?, premises imply conclusion?). Do not cancel factors without additional conditions.
View answer and explanation
(6, [23, 35], [47, 65], Yes, [1, 1], Yes, Yes)
Calculate both product remainders and keep premise truth, conclusion truth and material implication distinct. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [23, 35], [47, 65], Yes, [1, 1], Yes, Yes).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-mul-preservation · candidate
99. Multiplicative Congruence Values, Premises, and Material Implication
Use fixed positive modulus n=7, natural inputs a=24, b=45, c=49, d=56. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a*c) mod n,(b*d) mod n], product congruence?, premises imply conclusion?). Do not cancel factors without additional conditions.
View answer and explanation
(7, [24, 45], [49, 56], Yes, [0, 0], Yes, Yes)
Calculate both product remainders and keep premise truth, conclusion truth and material implication distinct. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (7, [24, 45], [49, 56], Yes, [0, 0], Yes, Yes).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-mul-preservation · candidate
100. Multiplicative Congruence Values, Premises, and Material Implication
Use fixed positive modulus n=8, natural inputs a=25, b=33, c=51, d=67. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a*c) mod n,(b*d) mod n], product congruence?, premises imply conclusion?). Do not cancel factors without additional conditions.
View answer and explanation
(8, [25, 33], [51, 67], Yes, [3, 3], Yes, Yes)
Calculate both product remainders and keep premise truth, conclusion truth and material implication distinct. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (8, [25, 33], [51, 67], Yes, [3, 3], Yes, Yes).
Reviewed step-by-step derivation
Natural-number congruence modulo a positive modulus · qf-ext18-modeq-mul-preservation · candidate
Canonical question JSONL · English JSONL · Validation report