數學問題集(49)---餘數問題

一個數被3除餘2，被5除餘3，被11除餘5，被17除餘7，那麼這個數最小是多少？

被3除餘2 = 被3除餘8

被5除餘3 = 被5除餘8

so 被3除餘2，被5除餘3 , the smallest is 8 , and every 28+ 15a ( a is an intenger . ) can 被3除餘2，被5除餘3 .

so , 被3除餘2，被5除餘3 = 被15除餘8

被11除餘5 = 被11除餘38

被15除餘8 = 被15除餘38

so ,被11除餘5 , 被15除餘8 , the smallest is 38

so , 被11除餘5 , 被15除餘8 = 被165除餘38

被17除餘7 and 被165除餘38 ,try :

165 x 1 + 38 = 203 ( 17 x 11 + 16 ) not 7

165 x 2 + 38 = 368 ( 17 x 21 + 11 ) not 7

165 x 3 + 38 = 533 ( 17 x 31 + 6 ) not 7

165 x 4 + 38 = 698 ( 17 x 41 + 1 ) not 7

165 x 5 + 38 = 863 ( 17 x 50 + 13 ) not 7

165 x 6 + 38 = 1028 ( 17 x 60 + 8 ) not 7

165 x 7 + 38 = 1193 ( 17 x 70 + 3 ) not 7

165 x 8 + 38 = 1358 ( 17 x 79 + 15 ) not 7

165 x 9 + 38 = 1523 ( 17 x 89 + 10 ) not 7

165 x 10 + 38 = 1688 ( 17 x 99 + 5 ) not 7

165 x 11 + 38 = 1853 ( 17 x 109 ) not 7

165 x 12 + 38 = 2018 ( 17 x 118 + 12 ) not 7

165 x 13 + 38 = 2183 ( 17 x 128 + 7 ) correct .

So , the answer is 2183

2010-09-24 16:19:46 補充：
自由自在 ,

Your way is OK . But your way spend more time and it is too diffcult , maybe F.2 or below don't know what you are writing .

Then I put it here:
http://img827.imageshack.us/img827/4463/67246550.png