※ 引述《andysleep (Maydayisgod)》之銘言：
: http://i.imgur.com/56mPsmu.jpg
: 有人會嗎？會用到柯西嗎 ？
: 感恩
123456789 = (11111 + a)(11111 - b)
= 123454321 + 11111(a - b) - ab
=> 2468 = 11111(a - b) - ab
(1) a,b 是自然數 => -ab < 0
=> 11111(a - b) > 0 => a - b > 0
(2) 如果 a - b 是奇數 => a,b為一奇一偶 => -ab 為偶數, 11111(a - b) 為奇數
=> 11111(a - b) - ab 為奇數 (2468是偶數) 矛盾
=> a - b 是偶數
(3) 如果 a - b 是 4m + 2 = 2(2m + 1) => a,b 為同奇或同偶
=>同奇 => 11111(a - b) 是偶數, -ab 是奇數,矛盾
=>同偶 => b = 2n => a = 4m + 2n + 2 = 2(2m + n + 1)
=> ab = 4n(2m + n + 1) 是4的倍數,11111(a - b)是2的倍數不是4的倍數
=> 11111(a - b) - ab 是2的倍數不是4的倍數 (2468是4的倍數)矛盾
=> a - b 是4的倍數
(4) a - b = 4, 8, 12, 16, 20
a - b = 4 => 2468 = 44444 - a(a-4) => a^2 - 4a - 41976 = 0
=> (a,b) = 無解
a - b = 8 => 2468 = 88888 - a(a-8) => a^2 - 8a - 86420 = 0
=> (a,b) = (298,290)
a - b =12 => 2468 = 133332 - a(a-12) => a^2 - 12a - 130864 = 0
=> (a,b) = 無解
a - b =16 => 2468 = 177776 - a(a-16) => a^2 - 16a - 175308 = 0
=> (a,b) = 無解
a - b =20 => 2468 = 222220 - a(a-20) => a^2 - 20a - 219752 = 0
=> (a,b) = 無解
(a,b) = (298,290) 唯一解
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