Each alphabet of KANGAROO is replaced with number by $2$ people; which alphabet is replaced with the same number?

In the word KANGAROO Bill and Bob replace the letters by digits, so that the resulting numbers are multiples of $11$. They each replace different letters by different digits and the same letters by the same digits ($K \not = 0$). Bill obtains the largest possible such number and Bob the smallest. In both cases one of the letters is replaced by the same digit. Which digit is this?

Please give me a hint on the answer or ways to approach this question.

$K+N+A+O=A+G+R+O+K11$
And so you can reduce to $K+N=G+R+K11$
You have that $K$ is different from $0$ so it must be $9$ in the maximum case and $1$ in the minimum one. Then to obtain the maximum you put $A=8$ and to have the minimum $A=0$, and so on...
• That's not quite right: the sum of the even and odd digits differs by a multiple of $11$. So you could have, for instance, $K=9, N=5, G=1, R=2$. – TonyK Feb 12 '16 at 10:33