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題目:兩正整數a,b的最大公因數大於1,且滿足[a,b]-(a,b)=253,求這兩個正整數. 答案:11,264或33,88或23,276或69,92或253,506 我只有答案沒有過程,我自己只找到(33,88)(69,92)(253,506) 請數學好手告訴我另外兩組答案怎麼求出來的?謝謝

設 (a,b) = d ==> a = md, b=nd 其中 (m,n) = 1 因此 [a, b] = mnd 又 [a, b] - (a, b) = 253 = mnd - d = d(mn - 1) 253 = 11*23 ==> d = 11, mn - 1 = 23 或 d = 23, mn - 1 = 11, d = 253, mn - 1 = 1 因此 d = 11, mn = 24 或 d = 23, mn = 12 或 d = 253, mn = 2 d = 11, mn = 24 ==> (m, n) = (1, 24), (2, 12), (3, 8), (4, 6) 紅色的不合( 因為要求 (m, n) = 1 ) d = 23, mn = 12 ==> (m, n) = (1, 12), (2, 6), (3, 4) d = 253, mn = 2 ==> (m, n) = (1, 2) 所以 d = 11, m = 1, n = 24 ==> a = 11, b = 264 d = 11, m = 3, n = 8 ==> a = 33, b = 88 d = 23, m = 1, n = 12 ==> a = 23, b = 276 d = 23, m = 3, n = 4 ==> a = 69, b = 92 d = 253, m = 1, n = 2 ==> a = 253, b = 506 共 (11, 264), (23, 276), (33, 88), (69, 92), (253, 506) 五組 嚴格來說 a, b 可以互換，所以有十組解

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