# Natural Divisibility, Explicit Witnesses, and Zero Boundaries

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

This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261013-advance-20261018-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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0001/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0002/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0003/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0004/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0005/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0006/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0007/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0008/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0009/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0010/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0011/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0012/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0013/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0014/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0015/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0016/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0017/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0018/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0019/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0020/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0021/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0022/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0023/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0024/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0025/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0026/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0027/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0028/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0029/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0030/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0031/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0032/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0033/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0034/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0035/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0036/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0037/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0038/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0039/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0040/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0041/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0042/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0043/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0044/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0045/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0046/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0047/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0048/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0049/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0050/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0051/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0052/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0053/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0054/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0055/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0056/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0057/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0058/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0059/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0060/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0061/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0062/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0063/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0064/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0065/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0066/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0067/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0068/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0069/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0070/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0071/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0072/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0073/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0074/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0075/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0076/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0077/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0078/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0079/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0080/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0081/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0082/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0083/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0084/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0085/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0086/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0087/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0088/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0089/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0090/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0091/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0092/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0093/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0094/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0095/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0096/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0097/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0098/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0099/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261018-r02-natural-divisibility-0100/)

Natural-number divisibility · qf-ext18-divisibility-transitivity · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261013-advance-20261018-r02-natural-divisibility-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261013-advance-20261018-r02-natural-divisibility-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261013-advance-20261018-r02-natural-divisibility-foundations/validation-report.json)