Re: [中學] 100年台南女中資優班

看板Math作者chz (稻草人騎士)時間12年前 (2013/07/15 18:22)推噓2(2推 0噓 1→)

留言3則, 2人參與討論串3/3 (看更多)

※ 引述《anous (阿文)》之銘言： : 1. 求介於5/8和8/13之間，分母最小且分子分母互質的分數。 : 　想請問這類型的問題有沒有比較有系統的處裡方法？ : 　我是硬找，雖然最後有找到但覺得沒什麼效率提供一個不用連分數的作法 Assume 8/13 < x/y < 5/8, and gcd(x,y) = 1 we have 13x - 8y > 0 and 8x - 5y < 0 since x and y is integer, assume 13x - 8y = k_1 > 0 and 8x-5y = k_2 < 0 where k_1 and k_2 is integer (13x - 8y)k_2 = k_1 k_2 = (8x - 5y)k_1 => (13 k_2 - 8 k_1) x - (8 k_2 - 5 k_1) y = 0 => x = (8 k_2 - 5 k_1), y = (13 k_2 - 8 k_1) gcd(8 k_2 - 5 k_1, 13 k_2 - 8 k_1) = gcd(8 k_2 - 5 k_1, 5 k_2 - 3 k_1) = gcd(3 k_2 - 2 k_1, 5 k_2 - 3 k_1) = gcd(3 k_2 - 2 k_1, 2 k_2 - 1 k_1) = gcd(1 k_2 , 2 k_2 - 1 k_1) = gcd(1 k_2 , 1 k_1) = gcd(x, y) = 1 since k_2 > 0, k_1 < 0, 13 k_2 - 8 k_1 has minimum value when k_1 = -1, k_2 = 1, that is x/y = (8+5)/(13+8)=13/21 -- ※ 發信站: 批踢踢實業坊(ptt.cc) ◆ From: 140.109.23.210

推

anous

07/16 00:03, , 1^F

07/16 00:03, 1^F

推

sckm160913

07/16 09:13, , 2^F

07/16 09:13, 2^F

→

sckm160913

07/16 09:13, , 3^F

07/16 09:13, 3^F