Answer:
[tex](x,y)=(4,4).[/tex]
Step-by-step explanation:
The given number 23x0255y7283 is divisible by both 9 and 11.
For any number to be divisible by 9, the sum of all the digit of the number must be divisible by 9.
Here, the sum of all the digits,
2+3+x+0+2+5+5+y+7+2+8+3=37+x+y.
The possible value of x and y are:
[tex]0\leq x\leq 9[/tex] and [tex]0\leq y\leq 9[/tex]
[tex]\Rightarrow 0\leq (x+y)\leq 18[/tex].
So, the possible value of (x+y) so that 37+x+y is divisible of 9 are:
x+y=8,17 ...(i)
Now. for any number to be divisible by 11, the difference of the sum of all the digits at the odd position and the sum of all the digits at the even position of the number must be either 0 or divisible by 11.
So, (2+x+2+5+7+8)-(3+0+5+y+2+3)=11+x-y is either 0 or divisible by 11.
As [tex]0\leq x\leq 9[/tex] and [tex]0\leq y\leq 9[/tex]
[tex]\Rightarrow 0\leq x-y\leq 9[/tex]
So, the possible value of (x-y) is 0.i.e
x-y=0
[tex]\Rightarrow x=y\;\cdots(iii)[/tex]
Form equation (i),
For x+y=8,
2x=8
[tex]\Rightarrow x=4[/tex].
So, from equation (iii), y=4. So
(x,y)=(4,4)
And for x+y=17,
2x=17
[tex]\Rightarrow x=8.5[/tex], which is not possible ay the digits x, y, are integer.
Hence, only one ordered pair (x,y) possible, which is
[tex](x,y)=(4,4).[/tex]
