4-digit number

[stuart clark]

(1) Two [tex]4[/tex]-digit numbers are to be formed such that the sum of the number is also a [tex]4[/tex]-digit number and in

no place the addition is with carrying the number of ways of forming the number under above conditions is

(2) a nunber in the form of [tex]xxyy[/tex] when divided by [tex]11[/tex] gives a number which is again divisible by [tex]11[/tex].

if [tex]x-y=1[/tex].Then the number is

[martin123456]

2.
[tex]xxyy=y+10y+100x+1000x=11y+1100x=11(y+100x)[/tex]
So [tex]xxyy:11=y+100x[/tex]
Since [tex]x-y=1\Rightarrow y=x-1\Rightarrow y+100x=101x-1=99x+2x-1\equiv 2x-1 \pmod{11}[/tex]
Since [tex]1\le 2x-1\le 17 \Rightarrow 2x-1=11\Rightarrow x=6\Rightarrow y=5[/tex]