由 訪客 於 星期四 十二月 09, 2010 10:18 am
n為整數,n被5,7,11除分別於4,3,5,求最小數為何?
n=5x+4=7y+3=11z+5
前二式5x+1=7y=5y+2y =>2y÷5餘1=>y÷5餘3 =>y=5k+3
後二式 7y=11z+2 =>35k+21=11z+2=>35k+19=11z=33k+11+(2k+8) => (2k+8)÷11餘0 =>2k÷11餘3=>k÷11餘7=>k=11m+7
=> y=5(11m+7)+3=55m+38
=>n=7(55m+38)+3=385m+269 ; m=0,1,2,... min(n)=269...ans