一條奧數題

考慮 OHDA0 + CCKO = HKMMO; 萬位O加進位後變成H
=> H = O + 1
H <> 0否則O為負數
H <> 1 否則 ABC * 1 = 0ABC = OHDA; A = C便重複了
以下是H = 2 至9的情況
H = 2, ABC * 2 = 12DA => 12DA/2 = 6XX => A = 6
6BC * 2 = 12D6 => C = 3 或8
C = 3則12DA0 + 33K1萬位無法進位
C = 8 則 ABC * M無法得到單數個位CCK1
H <> 2
H = 3, 23DA/3 = 7XX => A = 7
7BC * 3 = 23D7 => C = 9
23DA0 + 99K2 若百位有進千位變 33xxx百位不進千則變32xxx重複了2或3
H <> 3
H = 5, 45DA/5 = 9XX => A = 9
9BC * 5 = 45D9 無可能因 5乘任何數個位一定是0或5
H <> 5
H = 6, 56DA/6 = 9XX => A = 9
9BC * 6 = 56D9無可能因個位6是雙數9是單數
H<>6
H = 7, 67DA/7 = 9XX => A = 9
9BC * 7 = 78D9 => C = 7重複
H <> 7
H = 8, 78DA /8 = 9XX => A = 9
9BC * 8 = 78D9無可能因個位8是雙數9是單數
H <> 8
H = 9, 89DA/9 = 9XX => A = 9重複
H <> 9

圖片參考：http://img260.imageshack.us/img260/9311/13oct1.png

H = 4, O = 3, 34DA/4 = 8XX => A = 8
8BC * 4 = 34D8 => C = 2或7
C = 2則34DA0 + 22K3萬位無法進位變4KMM3
C = 7 => 8B7 * M = 77K3 => M = 9
34D80 + 77K3 = 4K993 => K = 1(留意十位)
34D80 + 7713 = 41993 => D = 2
3482 / 4 = 857 => B = 5

`````8 5 7
x `````4 9
------------
3 4 2 8
``7 7 1 3
------------
4 1 9 9 3

Hints: 先排列各digits的大小次序