A) Find the largest such number divisible by 11111 .
No matter what I try , I end up with atleast a digit repeating . Since the question says that the no. has all from 0 to 9 , therefore I cant solve it .
I think I need to find an algorithm for divisibility by 11111 to solve this . I cant find one .
B) How many no. of such numbers are possible that are divisible by 11111 ?
Our $10$-digit number $N$ is divisible by $11111$ if and only if it is divisible by $10^5-1$, for the digit sum of $N$ is divisible by $9$.
Let $a$ be the number whose decimal representation is given by the first $5$ digits of $N$, and $b$ the number whose decimal representation is given by the next $5$ digits of $N$. Then our $10$-digit number is divisible by $10^5-1$ if and only if $a+b$ is divisible by $10^5-1$. This is because $N=10^5a+b$, and $10^5\equiv 1\pmod{10^5-1}$.
Now that we know that $a+b$ must be divisible by $99999$, the rest should pose no difficulty.