Natural Divisibility, Explicit Witnesses, and Zero Boundaries
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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-r02-natural-divisibility-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. Natural Divisibility and an Explicit Multiplier Witness
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=0, b=0. Give (a, b, a divides b?, natural witness k for b=a*k or does-not-exist). For 0 divides 0 choose k=0, without claiming uniqueness.
View answer and explanation
(0, 0, Yes, 0)
Handle a=0 before division; otherwise require zero remainder and retain the exact natural quotient. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (0, 0, Yes, 0).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-witness · candidate
2. Natural Divisibility and an Explicit Multiplier Witness
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=0, b=7. Give (a, b, a divides b?, natural witness k for b=a*k or does-not-exist). For 0 divides 0 choose k=0, without claiming uniqueness.
View answer and explanation
(0, 7, No, does-not-exist)
Handle a=0 before division; otherwise require zero remainder and retain the exact natural quotient. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (0, 7, No, does-not-exist).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-witness · candidate
3. Natural Divisibility and an Explicit Multiplier Witness
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=2, b=7. Give (a, b, a divides b?, natural witness k for b=a*k or does-not-exist). For 0 divides 0 choose k=0, without claiming uniqueness.
View answer and explanation
(2, 7, No, does-not-exist)
Handle a=0 before division; otherwise require zero remainder and retain the exact natural quotient. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (2, 7, No, does-not-exist).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-witness · candidate
4. Natural Divisibility and an Explicit Multiplier Witness
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=3, b=12. Give (a, b, a divides b?, natural witness k for b=a*k or does-not-exist). For 0 divides 0 choose k=0, without claiming uniqueness.
View answer and explanation
(3, 12, Yes, 4)
Handle a=0 before division; otherwise require zero remainder and retain the exact natural quotient. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, 12, Yes, 4).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-witness · candidate
5. Natural Divisibility and an Explicit Multiplier Witness
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=4, b=21. Give (a, b, a divides b?, natural witness k for b=a*k or does-not-exist). For 0 divides 0 choose k=0, without claiming uniqueness.
View answer and explanation
(4, 21, No, does-not-exist)
Handle a=0 before division; otherwise require zero remainder and retain the exact natural quotient. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 21, No, does-not-exist).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-witness · candidate
6. Natural Divisibility and an Explicit Multiplier Witness
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=5, b=30. Give (a, b, a divides b?, natural witness k for b=a*k or does-not-exist). For 0 divides 0 choose k=0, without claiming uniqueness.
View answer and explanation
(5, 30, Yes, 6)
Handle a=0 before division; otherwise require zero remainder and retain the exact natural quotient. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, 30, Yes, 6).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-witness · candidate
7. Natural Divisibility and an Explicit Multiplier Witness
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=6, b=43. Give (a, b, a divides b?, natural witness k for b=a*k or does-not-exist). For 0 divides 0 choose k=0, without claiming uniqueness.
View answer and explanation
(6, 43, No, does-not-exist)
Handle a=0 before division; otherwise require zero remainder and retain the exact natural quotient. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, 43, No, does-not-exist).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-witness · candidate
8. Natural Divisibility and an Explicit Multiplier Witness
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=7, b=56. Give (a, b, a divides b?, natural witness k for b=a*k or does-not-exist). For 0 divides 0 choose k=0, without claiming uniqueness.
View answer and explanation
(7, 56, Yes, 8)
Handle a=0 before division; otherwise require zero remainder and retain the exact natural quotient. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (7, 56, Yes, 8).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-witness · candidate
9. Natural Divisibility and an Explicit Multiplier Witness
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=8, b=73. Give (a, b, a divides b?, natural witness k for b=a*k or does-not-exist). For 0 divides 0 choose k=0, without claiming uniqueness.
View answer and explanation
(8, 73, No, does-not-exist)
Handle a=0 before division; otherwise require zero remainder and retain the exact natural quotient. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (8, 73, No, does-not-exist).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-witness · candidate
10. Natural Divisibility and an Explicit Multiplier Witness
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=9, b=90. Give (a, b, a divides b?, natural witness k for b=a*k or does-not-exist). For 0 divides 0 choose k=0, without claiming uniqueness.
View answer and explanation
(9, 90, Yes, 10)
Handle a=0 before division; otherwise require zero remainder and retain the exact natural quotient. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (9, 90, Yes, 10).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-witness · candidate
11. Natural Divisibility and an Explicit Multiplier Witness
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=10, b=111. Give (a, b, a divides b?, natural witness k for b=a*k or does-not-exist). For 0 divides 0 choose k=0, without claiming uniqueness.
View answer and explanation
(10, 111, No, does-not-exist)
Handle a=0 before division; otherwise require zero remainder and retain the exact natural quotient. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (10, 111, No, does-not-exist).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-witness · candidate
12. Natural Divisibility and an Explicit Multiplier Witness
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=11, b=132. Give (a, b, a divides b?, natural witness k for b=a*k or does-not-exist). For 0 divides 0 choose k=0, without claiming uniqueness.
View answer and explanation
(11, 132, Yes, 12)
Handle a=0 before division; otherwise require zero remainder and retain the exact natural quotient. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (11, 132, Yes, 12).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-witness · candidate
13. Natural Divisibility and an Explicit Multiplier Witness
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=12, b=157. Give (a, b, a divides b?, natural witness k for b=a*k or does-not-exist). For 0 divides 0 choose k=0, without claiming uniqueness.
View answer and explanation
(12, 157, No, does-not-exist)
Handle a=0 before division; otherwise require zero remainder and retain the exact natural quotient. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (12, 157, No, does-not-exist).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-witness · candidate
14. Natural Divisibility and an Explicit Multiplier Witness
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=13, b=182. Give (a, b, a divides b?, natural witness k for b=a*k or does-not-exist). For 0 divides 0 choose k=0, without claiming uniqueness.
View answer and explanation
(13, 182, Yes, 14)
Handle a=0 before division; otherwise require zero remainder and retain the exact natural quotient. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (13, 182, Yes, 14).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-witness · candidate
15. Natural Divisibility and an Explicit Multiplier Witness
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=14, b=211. Give (a, b, a divides b?, natural witness k for b=a*k or does-not-exist). For 0 divides 0 choose k=0, without claiming uniqueness.
View answer and explanation
(14, 211, No, does-not-exist)
Handle a=0 before division; otherwise require zero remainder and retain the exact natural quotient. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (14, 211, No, does-not-exist).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-witness · candidate
16. Natural Divisibility and an Explicit Multiplier Witness
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=15, b=240. Give (a, b, a divides b?, natural witness k for b=a*k or does-not-exist). For 0 divides 0 choose k=0, without claiming uniqueness.
View answer and explanation
(15, 240, Yes, 16)
Handle a=0 before division; otherwise require zero remainder and retain the exact natural quotient. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (15, 240, Yes, 16).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-witness · candidate
17. Natural Divisibility and an Explicit Multiplier Witness
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=16, b=273. Give (a, b, a divides b?, natural witness k for b=a*k or does-not-exist). For 0 divides 0 choose k=0, without claiming uniqueness.
View answer and explanation
(16, 273, No, does-not-exist)
Handle a=0 before division; otherwise require zero remainder and retain the exact natural quotient. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (16, 273, No, does-not-exist).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-witness · candidate
18. Natural Divisibility and an Explicit Multiplier Witness
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=17, b=306. Give (a, b, a divides b?, natural witness k for b=a*k or does-not-exist). For 0 divides 0 choose k=0, without claiming uniqueness.
View answer and explanation
(17, 306, Yes, 18)
Handle a=0 before division; otherwise require zero remainder and retain the exact natural quotient. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (17, 306, Yes, 18).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-witness · candidate
19. Natural Divisibility and an Explicit Multiplier Witness
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=18, b=343. Give (a, b, a divides b?, natural witness k for b=a*k or does-not-exist). For 0 divides 0 choose k=0, without claiming uniqueness.
View answer and explanation
(18, 343, No, does-not-exist)
Handle a=0 before division; otherwise require zero remainder and retain the exact natural quotient. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (18, 343, No, does-not-exist).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-witness · candidate
20. Natural Divisibility and an Explicit Multiplier Witness
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=19, b=380. Give (a, b, a divides b?, natural witness k for b=a*k or does-not-exist). For 0 divides 0 choose k=0, without claiming uniqueness.
View answer and explanation
(19, 380, Yes, 20)
Handle a=0 before division; otherwise require zero remainder and retain the exact natural quotient. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (19, 380, Yes, 20).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-witness · candidate
21. Natural Divisibility and an Explicit Multiplier Witness
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=20, b=421. Give (a, b, a divides b?, natural witness k for b=a*k or does-not-exist). For 0 divides 0 choose k=0, without claiming uniqueness.
View answer and explanation
(20, 421, No, does-not-exist)
Handle a=0 before division; otherwise require zero remainder and retain the exact natural quotient. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (20, 421, No, does-not-exist).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-witness · candidate
22. Natural Divisibility and an Explicit Multiplier Witness
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=21, b=462. Give (a, b, a divides b?, natural witness k for b=a*k or does-not-exist). For 0 divides 0 choose k=0, without claiming uniqueness.
View answer and explanation
(21, 462, Yes, 22)
Handle a=0 before division; otherwise require zero remainder and retain the exact natural quotient. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (21, 462, Yes, 22).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-witness · candidate
23. Natural Divisibility and an Explicit Multiplier Witness
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=22, b=507. Give (a, b, a divides b?, natural witness k for b=a*k or does-not-exist). For 0 divides 0 choose k=0, without claiming uniqueness.
View answer and explanation
(22, 507, No, does-not-exist)
Handle a=0 before division; otherwise require zero remainder and retain the exact natural quotient. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (22, 507, No, does-not-exist).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-witness · candidate
24. Natural Divisibility and an Explicit Multiplier Witness
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=23, b=552. Give (a, b, a divides b?, natural witness k for b=a*k or does-not-exist). For 0 divides 0 choose k=0, without claiming uniqueness.
View answer and explanation
(23, 552, Yes, 24)
Handle a=0 before division; otherwise require zero remainder and retain the exact natural quotient. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (23, 552, Yes, 24).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-witness · candidate
25. Natural Divisibility and an Explicit Multiplier Witness
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=24, b=601. Give (a, b, a divides b?, natural witness k for b=a*k or does-not-exist). For 0 divides 0 choose k=0, without claiming uniqueness.
View answer and explanation
(24, 601, No, does-not-exist)
Handle a=0 before division; otherwise require zero remainder and retain the exact natural quotient. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (24, 601, No, does-not-exist).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-witness · candidate
26. Complete Positive Divisor Inventory Including the Zero Case
Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. b=0. Give (b, all positive divisors in increasing order or all-positive-naturals, count or infinite). This inventory excludes divisor 0 even when b=0.
View answer and explanation
(0, all-positive-naturals, infinite)
For a positive input enumerate every positive divisor; every positive natural divides zero, so its positive-divisor collection is infinite. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (0, all-positive-naturals, infinite).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-divisor-set · candidate
27. Complete Positive Divisor Inventory Including the Zero Case
Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. b=11. Give (b, all positive divisors in increasing order or all-positive-naturals, count or infinite). This inventory excludes divisor 0 even when b=0.
View answer and explanation
(11, [1, 11], 2)
For a positive input enumerate every positive divisor; every positive natural divides zero, so its positive-divisor collection is infinite. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (11, [1, 11], 2).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-divisor-set · candidate
28. Complete Positive Divisor Inventory Including the Zero Case
Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. b=12. Give (b, all positive divisors in increasing order or all-positive-naturals, count or infinite). This inventory excludes divisor 0 even when b=0.
View answer and explanation
(12, [1, 2, 3, 4, 6, 12], 6)
For a positive input enumerate every positive divisor; every positive natural divides zero, so its positive-divisor collection is infinite. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (12, [1, 2, 3, 4, 6, 12], 6).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-divisor-set · candidate
29. Complete Positive Divisor Inventory Including the Zero Case
Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. b=13. Give (b, all positive divisors in increasing order or all-positive-naturals, count or infinite). This inventory excludes divisor 0 even when b=0.
View answer and explanation
(13, [1, 13], 2)
For a positive input enumerate every positive divisor; every positive natural divides zero, so its positive-divisor collection is infinite. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (13, [1, 13], 2).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-divisor-set · candidate
30. Complete Positive Divisor Inventory Including the Zero Case
Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. b=14. Give (b, all positive divisors in increasing order or all-positive-naturals, count or infinite). This inventory excludes divisor 0 even when b=0.
View answer and explanation
(14, [1, 2, 7, 14], 4)
For a positive input enumerate every positive divisor; every positive natural divides zero, so its positive-divisor collection is infinite. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (14, [1, 2, 7, 14], 4).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-divisor-set · candidate
31. Complete Positive Divisor Inventory Including the Zero Case
Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. b=15. Give (b, all positive divisors in increasing order or all-positive-naturals, count or infinite). This inventory excludes divisor 0 even when b=0.
View answer and explanation
(15, [1, 3, 5, 15], 4)
For a positive input enumerate every positive divisor; every positive natural divides zero, so its positive-divisor collection is infinite. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (15, [1, 3, 5, 15], 4).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-divisor-set · candidate
32. Complete Positive Divisor Inventory Including the Zero Case
Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. b=16. Give (b, all positive divisors in increasing order or all-positive-naturals, count or infinite). This inventory excludes divisor 0 even when b=0.
View answer and explanation
(16, [1, 2, 4, 8, 16], 5)
For a positive input enumerate every positive divisor; every positive natural divides zero, so its positive-divisor collection is infinite. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (16, [1, 2, 4, 8, 16], 5).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-divisor-set · candidate
33. Complete Positive Divisor Inventory Including the Zero Case
Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. b=17. Give (b, all positive divisors in increasing order or all-positive-naturals, count or infinite). This inventory excludes divisor 0 even when b=0.
View answer and explanation
(17, [1, 17], 2)
For a positive input enumerate every positive divisor; every positive natural divides zero, so its positive-divisor collection is infinite. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (17, [1, 17], 2).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-divisor-set · candidate
34. Complete Positive Divisor Inventory Including the Zero Case
Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. b=18. Give (b, all positive divisors in increasing order or all-positive-naturals, count or infinite). This inventory excludes divisor 0 even when b=0.
View answer and explanation
(18, [1, 2, 3, 6, 9, 18], 6)
For a positive input enumerate every positive divisor; every positive natural divides zero, so its positive-divisor collection is infinite. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (18, [1, 2, 3, 6, 9, 18], 6).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-divisor-set · candidate
35. Complete Positive Divisor Inventory Including the Zero Case
Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. b=19. Give (b, all positive divisors in increasing order or all-positive-naturals, count or infinite). This inventory excludes divisor 0 even when b=0.
View answer and explanation
(19, [1, 19], 2)
For a positive input enumerate every positive divisor; every positive natural divides zero, so its positive-divisor collection is infinite. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (19, [1, 19], 2).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-divisor-set · candidate
36. Complete Positive Divisor Inventory Including the Zero Case
Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. b=20. Give (b, all positive divisors in increasing order or all-positive-naturals, count or infinite). This inventory excludes divisor 0 even when b=0.
View answer and explanation
(20, [1, 2, 4, 5, 10, 20], 6)
For a positive input enumerate every positive divisor; every positive natural divides zero, so its positive-divisor collection is infinite. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (20, [1, 2, 4, 5, 10, 20], 6).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-divisor-set · candidate
37. Complete Positive Divisor Inventory Including the Zero Case
Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. b=21. Give (b, all positive divisors in increasing order or all-positive-naturals, count or infinite). This inventory excludes divisor 0 even when b=0.
View answer and explanation
(21, [1, 3, 7, 21], 4)
For a positive input enumerate every positive divisor; every positive natural divides zero, so its positive-divisor collection is infinite. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (21, [1, 3, 7, 21], 4).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-divisor-set · candidate
38. Complete Positive Divisor Inventory Including the Zero Case
Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. b=22. Give (b, all positive divisors in increasing order or all-positive-naturals, count or infinite). This inventory excludes divisor 0 even when b=0.
View answer and explanation
(22, [1, 2, 11, 22], 4)
For a positive input enumerate every positive divisor; every positive natural divides zero, so its positive-divisor collection is infinite. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (22, [1, 2, 11, 22], 4).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-divisor-set · candidate
39. Complete Positive Divisor Inventory Including the Zero Case
Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. b=23. Give (b, all positive divisors in increasing order or all-positive-naturals, count or infinite). This inventory excludes divisor 0 even when b=0.
View answer and explanation
(23, [1, 23], 2)
For a positive input enumerate every positive divisor; every positive natural divides zero, so its positive-divisor collection is infinite. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (23, [1, 23], 2).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-divisor-set · candidate
40. Complete Positive Divisor Inventory Including the Zero Case
Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. b=24. Give (b, all positive divisors in increasing order or all-positive-naturals, count or infinite). This inventory excludes divisor 0 even when b=0.
View answer and explanation
(24, [1, 2, 3, 4, 6, 8, 12, 24], 8)
For a positive input enumerate every positive divisor; every positive natural divides zero, so its positive-divisor collection is infinite. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (24, [1, 2, 3, 4, 6, 8, 12, 24], 8).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-divisor-set · candidate
41. Complete Positive Divisor Inventory Including the Zero Case
Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. b=25. Give (b, all positive divisors in increasing order or all-positive-naturals, count or infinite). This inventory excludes divisor 0 even when b=0.
View answer and explanation
(25, [1, 5, 25], 3)
For a positive input enumerate every positive divisor; every positive natural divides zero, so its positive-divisor collection is infinite. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (25, [1, 5, 25], 3).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-divisor-set · candidate
42. Complete Positive Divisor Inventory Including the Zero Case
Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. b=26. Give (b, all positive divisors in increasing order or all-positive-naturals, count or infinite). This inventory excludes divisor 0 even when b=0.
View answer and explanation
(26, [1, 2, 13, 26], 4)
For a positive input enumerate every positive divisor; every positive natural divides zero, so its positive-divisor collection is infinite. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (26, [1, 2, 13, 26], 4).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-divisor-set · candidate
43. Complete Positive Divisor Inventory Including the Zero Case
Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. b=27. Give (b, all positive divisors in increasing order or all-positive-naturals, count or infinite). This inventory excludes divisor 0 even when b=0.
View answer and explanation
(27, [1, 3, 9, 27], 4)
For a positive input enumerate every positive divisor; every positive natural divides zero, so its positive-divisor collection is infinite. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (27, [1, 3, 9, 27], 4).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-divisor-set · candidate
44. Complete Positive Divisor Inventory Including the Zero Case
Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. b=28. Give (b, all positive divisors in increasing order or all-positive-naturals, count or infinite). This inventory excludes divisor 0 even when b=0.
View answer and explanation
(28, [1, 2, 4, 7, 14, 28], 6)
For a positive input enumerate every positive divisor; every positive natural divides zero, so its positive-divisor collection is infinite. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (28, [1, 2, 4, 7, 14, 28], 6).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-divisor-set · candidate
45. Complete Positive Divisor Inventory Including the Zero Case
Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. b=29. Give (b, all positive divisors in increasing order or all-positive-naturals, count or infinite). This inventory excludes divisor 0 even when b=0.
View answer and explanation
(29, [1, 29], 2)
For a positive input enumerate every positive divisor; every positive natural divides zero, so its positive-divisor collection is infinite. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (29, [1, 29], 2).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-divisor-set · candidate
46. Complete Positive Divisor Inventory Including the Zero Case
Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. b=30. Give (b, all positive divisors in increasing order or all-positive-naturals, count or infinite). This inventory excludes divisor 0 even when b=0.
View answer and explanation
(30, [1, 2, 3, 5, 6, 10, 15, 30], 8)
For a positive input enumerate every positive divisor; every positive natural divides zero, so its positive-divisor collection is infinite. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (30, [1, 2, 3, 5, 6, 10, 15, 30], 8).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-divisor-set · candidate
47. Complete Positive Divisor Inventory Including the Zero Case
Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. b=31. Give (b, all positive divisors in increasing order or all-positive-naturals, count or infinite). This inventory excludes divisor 0 even when b=0.
View answer and explanation
(31, [1, 31], 2)
For a positive input enumerate every positive divisor; every positive natural divides zero, so its positive-divisor collection is infinite. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (31, [1, 31], 2).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-divisor-set · candidate
48. Complete Positive Divisor Inventory Including the Zero Case
Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. b=32. Give (b, all positive divisors in increasing order or all-positive-naturals, count or infinite). This inventory excludes divisor 0 even when b=0.
View answer and explanation
(32, [1, 2, 4, 8, 16, 32], 6)
For a positive input enumerate every positive divisor; every positive natural divides zero, so its positive-divisor collection is infinite. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (32, [1, 2, 4, 8, 16, 32], 6).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-divisor-set · candidate
49. Complete Positive Divisor Inventory Including the Zero Case
Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. b=33. Give (b, all positive divisors in increasing order or all-positive-naturals, count or infinite). This inventory excludes divisor 0 even when b=0.
View answer and explanation
(33, [1, 3, 11, 33], 4)
For a positive input enumerate every positive divisor; every positive natural divides zero, so its positive-divisor collection is infinite. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (33, [1, 3, 11, 33], 4).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-divisor-set · candidate
50. Complete Positive Divisor Inventory Including the Zero Case
Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. b=34. Give (b, all positive divisors in increasing order or all-positive-naturals, count or infinite). This inventory excludes divisor 0 even when b=0.
View answer and explanation
(34, [1, 2, 17, 34], 4)
For a positive input enumerate every positive divisor; every positive natural divides zero, so its positive-divisor collection is infinite. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (34, [1, 2, 17, 34], 4).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-divisor-set · candidate
51. All Common Multiples in an Inclusive Natural Window
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=0, b=0, bound=30. Give (a, b, bound, all common multiples in [0,bound] in increasing order).
View answer and explanation
(0, 0, 30, [0])
Inspect the complete inclusive window and apply both divisibility guards; zero is retained when appropriate. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (0, 0, 30, [0]).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-common-multiples · candidate
52. All Common Multiples in an Inclusive Natural Window
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=1, b=2, bound=31. Give (a, b, bound, all common multiples in [0,bound] in increasing order).
View answer and explanation
(1, 2, 31, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30])
Inspect the complete inclusive window and apply both divisibility guards; zero is retained when appropriate. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (1, 2, 31, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30]).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-common-multiples · candidate
53. All Common Multiples in an Inclusive Natural Window
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=2, b=4, bound=32. Give (a, b, bound, all common multiples in [0,bound] in increasing order).
View answer and explanation
(2, 4, 32, [0, 4, 8, 12, 16, 20, 24, 28, 32])
Inspect the complete inclusive window and apply both divisibility guards; zero is retained when appropriate. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (2, 4, 32, [0, 4, 8, 12, 16, 20, 24, 28, 32]).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-common-multiples · candidate
54. All Common Multiples in an Inclusive Natural Window
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=3, b=6, bound=33. Give (a, b, bound, all common multiples in [0,bound] in increasing order).
View answer and explanation
(3, 6, 33, [0, 6, 12, 18, 24, 30])
Inspect the complete inclusive window and apply both divisibility guards; zero is retained when appropriate. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, 6, 33, [0, 6, 12, 18, 24, 30]).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-common-multiples · candidate
55. All Common Multiples in an Inclusive Natural Window
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=4, b=8, bound=34. Give (a, b, bound, all common multiples in [0,bound] in increasing order).
View answer and explanation
(4, 8, 34, [0, 8, 16, 24, 32])
Inspect the complete inclusive window and apply both divisibility guards; zero is retained when appropriate. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 8, 34, [0, 8, 16, 24, 32]).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-common-multiples · candidate
56. All Common Multiples in an Inclusive Natural Window
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=5, b=1, bound=35. Give (a, b, bound, all common multiples in [0,bound] in increasing order).
View answer and explanation
(5, 1, 35, [0, 5, 10, 15, 20, 25, 30, 35])
Inspect the complete inclusive window and apply both divisibility guards; zero is retained when appropriate. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, 1, 35, [0, 5, 10, 15, 20, 25, 30, 35]).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-common-multiples · candidate
57. All Common Multiples in an Inclusive Natural Window
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=6, b=3, bound=36. Give (a, b, bound, all common multiples in [0,bound] in increasing order).
View answer and explanation
(6, 3, 36, [0, 6, 12, 18, 24, 30, 36])
Inspect the complete inclusive window and apply both divisibility guards; zero is retained when appropriate. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, 3, 36, [0, 6, 12, 18, 24, 30, 36]).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-common-multiples · candidate
58. All Common Multiples in an Inclusive Natural Window
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=0, b=5, bound=37. Give (a, b, bound, all common multiples in [0,bound] in increasing order).
View answer and explanation
(0, 5, 37, [0])
Inspect the complete inclusive window and apply both divisibility guards; zero is retained when appropriate. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (0, 5, 37, [0]).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-common-multiples · candidate
59. All Common Multiples in an Inclusive Natural Window
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=1, b=7, bound=38. Give (a, b, bound, all common multiples in [0,bound] in increasing order).
View answer and explanation
(1, 7, 38, [0, 7, 14, 21, 28, 35])
Inspect the complete inclusive window and apply both divisibility guards; zero is retained when appropriate. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (1, 7, 38, [0, 7, 14, 21, 28, 35]).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-common-multiples · candidate
60. All Common Multiples in an Inclusive Natural Window
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=2, b=0, bound=39. Give (a, b, bound, all common multiples in [0,bound] in increasing order).
View answer and explanation
(2, 0, 39, [0])
Inspect the complete inclusive window and apply both divisibility guards; zero is retained when appropriate. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (2, 0, 39, [0]).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-common-multiples · candidate
61. All Common Multiples in an Inclusive Natural Window
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=3, b=2, bound=40. Give (a, b, bound, all common multiples in [0,bound] in increasing order).
View answer and explanation
(3, 2, 40, [0, 6, 12, 18, 24, 30, 36])
Inspect the complete inclusive window and apply both divisibility guards; zero is retained when appropriate. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, 2, 40, [0, 6, 12, 18, 24, 30, 36]).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-common-multiples · candidate
62. All Common Multiples in an Inclusive Natural Window
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=4, b=4, bound=41. Give (a, b, bound, all common multiples in [0,bound] in increasing order).
View answer and explanation
(4, 4, 41, [0, 4, 8, 12, 16, 20, 24, 28, 32, 36, 40])
Inspect the complete inclusive window and apply both divisibility guards; zero is retained when appropriate. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 4, 41, [0, 4, 8, 12, 16, 20, 24, 28, 32, 36, 40]).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-common-multiples · candidate
63. All Common Multiples in an Inclusive Natural Window
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=5, b=6, bound=42. Give (a, b, bound, all common multiples in [0,bound] in increasing order).
View answer and explanation
(5, 6, 42, [0, 30])
Inspect the complete inclusive window and apply both divisibility guards; zero is retained when appropriate. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, 6, 42, [0, 30]).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-common-multiples · candidate
64. All Common Multiples in an Inclusive Natural Window
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=6, b=8, bound=43. Give (a, b, bound, all common multiples in [0,bound] in increasing order).
View answer and explanation
(6, 8, 43, [0, 24])
Inspect the complete inclusive window and apply both divisibility guards; zero is retained when appropriate. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, 8, 43, [0, 24]).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-common-multiples · candidate
65. All Common Multiples in an Inclusive Natural Window
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=0, b=1, bound=44. Give (a, b, bound, all common multiples in [0,bound] in increasing order).
View answer and explanation
(0, 1, 44, [0])
Inspect the complete inclusive window and apply both divisibility guards; zero is retained when appropriate. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (0, 1, 44, [0]).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-common-multiples · candidate
66. All Common Multiples in an Inclusive Natural Window
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=1, b=3, bound=45. Give (a, b, bound, all common multiples in [0,bound] in increasing order).
View answer and explanation
(1, 3, 45, [0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45])
Inspect the complete inclusive window and apply both divisibility guards; zero is retained when appropriate. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (1, 3, 45, [0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45]).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-common-multiples · candidate
67. All Common Multiples in an Inclusive Natural Window
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=2, b=5, bound=46. Give (a, b, bound, all common multiples in [0,bound] in increasing order).
View answer and explanation
(2, 5, 46, [0, 10, 20, 30, 40])
Inspect the complete inclusive window and apply both divisibility guards; zero is retained when appropriate. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (2, 5, 46, [0, 10, 20, 30, 40]).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-common-multiples · candidate
68. All Common Multiples in an Inclusive Natural Window
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=3, b=7, bound=47. Give (a, b, bound, all common multiples in [0,bound] in increasing order).
View answer and explanation
(3, 7, 47, [0, 21, 42])
Inspect the complete inclusive window and apply both divisibility guards; zero is retained when appropriate. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, 7, 47, [0, 21, 42]).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-common-multiples · candidate
69. All Common Multiples in an Inclusive Natural Window
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=4, b=0, bound=48. Give (a, b, bound, all common multiples in [0,bound] in increasing order).
View answer and explanation
(4, 0, 48, [0])
Inspect the complete inclusive window and apply both divisibility guards; zero is retained when appropriate. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 0, 48, [0]).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-common-multiples · candidate
70. All Common Multiples in an Inclusive Natural Window
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=5, b=2, bound=49. Give (a, b, bound, all common multiples in [0,bound] in increasing order).
View answer and explanation
(5, 2, 49, [0, 10, 20, 30, 40])
Inspect the complete inclusive window and apply both divisibility guards; zero is retained when appropriate. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, 2, 49, [0, 10, 20, 30, 40]).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-common-multiples · candidate
71. All Common Multiples in an Inclusive Natural Window
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=6, b=4, bound=50. Give (a, b, bound, all common multiples in [0,bound] in increasing order).
View answer and explanation
(6, 4, 50, [0, 12, 24, 36, 48])
Inspect the complete inclusive window and apply both divisibility guards; zero is retained when appropriate. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, 4, 50, [0, 12, 24, 36, 48]).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-common-multiples · candidate
72. All Common Multiples in an Inclusive Natural Window
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=0, b=6, bound=51. Give (a, b, bound, all common multiples in [0,bound] in increasing order).
View answer and explanation
(0, 6, 51, [0])
Inspect the complete inclusive window and apply both divisibility guards; zero is retained when appropriate. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (0, 6, 51, [0]).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-common-multiples · candidate
73. All Common Multiples in an Inclusive Natural Window
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=1, b=8, bound=52. Give (a, b, bound, all common multiples in [0,bound] in increasing order).
View answer and explanation
(1, 8, 52, [0, 8, 16, 24, 32, 40, 48])
Inspect the complete inclusive window and apply both divisibility guards; zero is retained when appropriate. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (1, 8, 52, [0, 8, 16, 24, 32, 40, 48]).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-common-multiples · candidate
74. All Common Multiples in an Inclusive Natural Window
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=2, b=1, bound=53. Give (a, b, bound, all common multiples in [0,bound] in increasing order).
View answer and explanation
(2, 1, 53, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52])
Inspect the complete inclusive window and apply both divisibility guards; zero is retained when appropriate. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (2, 1, 53, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52]).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-common-multiples · candidate
75. All Common Multiples in an Inclusive Natural Window
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=3, b=3, bound=54. Give (a, b, bound, all common multiples in [0,bound] in increasing order).
View answer and explanation
(3, 3, 54, [0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54])
Inspect the complete inclusive window and apply both divisibility guards; zero is retained when appropriate. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, 3, 54, [0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54]).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-common-multiples · candidate
76. Compose Two Given Natural Divisibility Witnesses
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=0, k=0, l=0; define b=a*k and c=b*l. Give (a, b, c, k, l, k*l, a divides c?).
View answer and explanation
(0, 0, 0, 0, 0, 0, Yes)
Multiply the explicit witnesses in the given order and check the resulting exact equality, including zero cases. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (0, 0, 0, 0, 0, 0, Yes).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-transitivity · candidate
77. Compose Two Given Natural Divisibility Witnesses
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=1, k=1, l=3; define b=a*k and c=b*l. Give (a, b, c, k, l, k*l, a divides c?).
View answer and explanation
(1, 1, 3, 1, 3, 3, Yes)
Multiply the explicit witnesses in the given order and check the resulting exact equality, including zero cases. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (1, 1, 3, 1, 3, 3, Yes).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-transitivity · candidate
78. Compose Two Given Natural Divisibility Witnesses
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=2, k=2, l=6; define b=a*k and c=b*l. Give (a, b, c, k, l, k*l, a divides c?).
View answer and explanation
(2, 4, 24, 2, 6, 12, Yes)
Multiply the explicit witnesses in the given order and check the resulting exact equality, including zero cases. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (2, 4, 24, 2, 6, 12, Yes).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-transitivity · candidate
79. Compose Two Given Natural Divisibility Witnesses
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=3, k=3, l=2; define b=a*k and c=b*l. Give (a, b, c, k, l, k*l, a divides c?).
View answer and explanation
(3, 9, 18, 3, 2, 6, Yes)
Multiply the explicit witnesses in the given order and check the resulting exact equality, including zero cases. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, 9, 18, 3, 2, 6, Yes).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-transitivity · candidate
80. Compose Two Given Natural Divisibility Witnesses
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=4, k=4, l=5; define b=a*k and c=b*l. Give (a, b, c, k, l, k*l, a divides c?).
View answer and explanation
(4, 16, 80, 4, 5, 20, Yes)
Multiply the explicit witnesses in the given order and check the resulting exact equality, including zero cases. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 16, 80, 4, 5, 20, Yes).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-transitivity · candidate
81. Compose Two Given Natural Divisibility Witnesses
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=5, k=0, l=1; define b=a*k and c=b*l. Give (a, b, c, k, l, k*l, a divides c?).
View answer and explanation
(5, 0, 0, 0, 1, 0, Yes)
Multiply the explicit witnesses in the given order and check the resulting exact equality, including zero cases. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, 0, 0, 0, 1, 0, Yes).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-transitivity · candidate
82. Compose Two Given Natural Divisibility Witnesses
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=0, k=1, l=4; define b=a*k and c=b*l. Give (a, b, c, k, l, k*l, a divides c?).
View answer and explanation
(0, 0, 0, 1, 4, 4, Yes)
Multiply the explicit witnesses in the given order and check the resulting exact equality, including zero cases. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (0, 0, 0, 1, 4, 4, Yes).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-transitivity · candidate
83. Compose Two Given Natural Divisibility Witnesses
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=1, k=2, l=0; define b=a*k and c=b*l. Give (a, b, c, k, l, k*l, a divides c?).
View answer and explanation
(1, 2, 0, 2, 0, 0, Yes)
Multiply the explicit witnesses in the given order and check the resulting exact equality, including zero cases. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (1, 2, 0, 2, 0, 0, Yes).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-transitivity · candidate
84. Compose Two Given Natural Divisibility Witnesses
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=2, k=3, l=3; define b=a*k and c=b*l. Give (a, b, c, k, l, k*l, a divides c?).
View answer and explanation
(2, 6, 18, 3, 3, 9, Yes)
Multiply the explicit witnesses in the given order and check the resulting exact equality, including zero cases. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (2, 6, 18, 3, 3, 9, Yes).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-transitivity · candidate
85. Compose Two Given Natural Divisibility Witnesses
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=3, k=4, l=6; define b=a*k and c=b*l. Give (a, b, c, k, l, k*l, a divides c?).
View answer and explanation
(3, 12, 72, 4, 6, 24, Yes)
Multiply the explicit witnesses in the given order and check the resulting exact equality, including zero cases. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, 12, 72, 4, 6, 24, Yes).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-transitivity · candidate
86. Compose Two Given Natural Divisibility Witnesses
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=4, k=0, l=2; define b=a*k and c=b*l. Give (a, b, c, k, l, k*l, a divides c?).
View answer and explanation
(4, 0, 0, 0, 2, 0, Yes)
Multiply the explicit witnesses in the given order and check the resulting exact equality, including zero cases. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 0, 0, 0, 2, 0, Yes).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-transitivity · candidate
87. Compose Two Given Natural Divisibility Witnesses
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=5, k=1, l=5; define b=a*k and c=b*l. Give (a, b, c, k, l, k*l, a divides c?).
View answer and explanation
(5, 5, 25, 1, 5, 5, Yes)
Multiply the explicit witnesses in the given order and check the resulting exact equality, including zero cases. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, 5, 25, 1, 5, 5, Yes).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-transitivity · candidate
88. Compose Two Given Natural Divisibility Witnesses
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=0, k=2, l=1; define b=a*k and c=b*l. Give (a, b, c, k, l, k*l, a divides c?).
View answer and explanation
(0, 0, 0, 2, 1, 2, Yes)
Multiply the explicit witnesses in the given order and check the resulting exact equality, including zero cases. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (0, 0, 0, 2, 1, 2, Yes).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-transitivity · candidate
89. Compose Two Given Natural Divisibility Witnesses
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=1, k=3, l=4; define b=a*k and c=b*l. Give (a, b, c, k, l, k*l, a divides c?).
View answer and explanation
(1, 3, 12, 3, 4, 12, Yes)
Multiply the explicit witnesses in the given order and check the resulting exact equality, including zero cases. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (1, 3, 12, 3, 4, 12, Yes).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-transitivity · candidate
90. Compose Two Given Natural Divisibility Witnesses
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=2, k=4, l=0; define b=a*k and c=b*l. Give (a, b, c, k, l, k*l, a divides c?).
View answer and explanation
(2, 8, 0, 4, 0, 0, Yes)
Multiply the explicit witnesses in the given order and check the resulting exact equality, including zero cases. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (2, 8, 0, 4, 0, 0, Yes).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-transitivity · candidate
91. Compose Two Given Natural Divisibility Witnesses
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=3, k=0, l=3; define b=a*k and c=b*l. Give (a, b, c, k, l, k*l, a divides c?).
View answer and explanation
(3, 0, 0, 0, 3, 0, Yes)
Multiply the explicit witnesses in the given order and check the resulting exact equality, including zero cases. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, 0, 0, 0, 3, 0, Yes).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-transitivity · candidate
92. Compose Two Given Natural Divisibility Witnesses
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=4, k=1, l=6; define b=a*k and c=b*l. Give (a, b, c, k, l, k*l, a divides c?).
View answer and explanation
(4, 4, 24, 1, 6, 6, Yes)
Multiply the explicit witnesses in the given order and check the resulting exact equality, including zero cases. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 4, 24, 1, 6, 6, Yes).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-transitivity · candidate
93. Compose Two Given Natural Divisibility Witnesses
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=5, k=2, l=2; define b=a*k and c=b*l. Give (a, b, c, k, l, k*l, a divides c?).
View answer and explanation
(5, 10, 20, 2, 2, 4, Yes)
Multiply the explicit witnesses in the given order and check the resulting exact equality, including zero cases. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, 10, 20, 2, 2, 4, Yes).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-transitivity · candidate
94. Compose Two Given Natural Divisibility Witnesses
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=0, k=3, l=5; define b=a*k and c=b*l. Give (a, b, c, k, l, k*l, a divides c?).
View answer and explanation
(0, 0, 0, 3, 5, 15, Yes)
Multiply the explicit witnesses in the given order and check the resulting exact equality, including zero cases. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (0, 0, 0, 3, 5, 15, Yes).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-transitivity · candidate
95. Compose Two Given Natural Divisibility Witnesses
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=1, k=4, l=1; define b=a*k and c=b*l. Give (a, b, c, k, l, k*l, a divides c?).
View answer and explanation
(1, 4, 4, 4, 1, 4, Yes)
Multiply the explicit witnesses in the given order and check the resulting exact equality, including zero cases. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (1, 4, 4, 4, 1, 4, Yes).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-transitivity · candidate
96. Compose Two Given Natural Divisibility Witnesses
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=2, k=0, l=4; define b=a*k and c=b*l. Give (a, b, c, k, l, k*l, a divides c?).
View answer and explanation
(2, 0, 0, 0, 4, 0, Yes)
Multiply the explicit witnesses in the given order and check the resulting exact equality, including zero cases. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (2, 0, 0, 0, 4, 0, Yes).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-transitivity · candidate
97. Compose Two Given Natural Divisibility Witnesses
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=3, k=1, l=0; define b=a*k and c=b*l. Give (a, b, c, k, l, k*l, a divides c?).
View answer and explanation
(3, 3, 0, 1, 0, 0, Yes)
Multiply the explicit witnesses in the given order and check the resulting exact equality, including zero cases. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, 3, 0, 1, 0, 0, Yes).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-transitivity · candidate
98. Compose Two Given Natural Divisibility Witnesses
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=4, k=2, l=3; define b=a*k and c=b*l. Give (a, b, c, k, l, k*l, a divides c?).
View answer and explanation
(4, 8, 24, 2, 3, 6, Yes)
Multiply the explicit witnesses in the given order and check the resulting exact equality, including zero cases. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 8, 24, 2, 3, 6, Yes).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-transitivity · candidate
99. Compose Two Given Natural Divisibility Witnesses
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=5, k=3, l=6; define b=a*k and c=b*l. Give (a, b, c, k, l, k*l, a divides c?).
View answer and explanation
(5, 15, 90, 3, 6, 18, Yes)
Multiply the explicit witnesses in the given order and check the resulting exact equality, including zero cases. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, 15, 90, 3, 6, 18, Yes).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-transitivity · candidate
100. Compose Two Given Natural Divisibility Witnesses
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=0, k=4, l=2; define b=a*k and c=b*l. Give (a, b, c, k, l, k*l, a divides c?).
View answer and explanation
(0, 0, 0, 4, 2, 8, Yes)
Multiply the explicit witnesses in the given order and check the resulting exact equality, including zero cases. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (0, 0, 0, 4, 2, 8, Yes).
Reviewed step-by-step derivation
Natural-number divisibility · qf-ext18-divisibility-transitivity · candidate
Canonical question JSONL · English JSONL · Validation report