{"schema_version":"ocme-site-results-v1","locale":"en","page_type":"derivations","facts":{"questions":36000,"reviewed":36000,"display_candidates":0,"knowledge":302,"guides":299,"paths":59,"batches":360,"families":1441,"atlas":302,"core":80,"evidence":9,"formal_evidence":5,"formal_subjects":4,"lean_files":6,"lean_toolchain":"leanprover/lean4:v4.30.0","question_state":"candidate","english_state":"ai_candidate","question_Lean":"NOT_RUN","canonical_truth_promotion":false},"filters":{"q":"mko-natural-divisibility"},"kind":"guides","total":1,"pages":1,"page":1,"entries":[{"id":"derivation-mko-natural-divisibility-01","title":{"zh-Hant":"自然數整除：以乘數見證判定、處理 0 的情形並證明遞移性","en":"Natural-number divisibility: deciding by a multiplier witness, handling zero, and proving transitivity"},"summary":{"zh-Hant":"依 MKO 定義（a∣b ⇔ 存在自然數 k 使 b=a·k）：(1) 證明 6∣84；(2) 證明 6∤87；(3) 證明 4∤10，雖然 10/4=5/2 是一個數；(4) 證明每個 a 都整除 0；(5) 證明 0∣b ⇔ b=0，並說明 0∣0 的見證不唯一、0∤5；(6) 證明 a>0 時見證至多一個（若 a∣b，則既有見證唯一）；(7) 證明整除的自反性與遞移性，並由 3∣6、6∣84 求出 3∣84 的見證。","en":"From the MKO's definition (a∣b ⇔ there is a natural k with b=a·k): (1) prove 6∣84; (2) prove 6∤87; (3) prove 4∤10, although 10/4=5/2 is a number; (4) prove that every a divides 0; (5) prove 0∣b ⇔ b=0, and explain why the witness for 0∣0 is not unique and why 0∤5; (6) prove that for a>0 there is at most one witness (and when a∣b that existing witness is unique); (7) prove reflexivity and transitivity of divisibility, and from 3∣6 and 6∣84 find a witness for 3∣84."},"mko_ids":["mko-natural-divisibility"]}],"title":"Concept derivation guides"}