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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.

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You have to know that in the multiples of 11 the sum of even digits and odd digits is the same or differ for a multiple of 11. So,you can write in both the cases:

$K+N+A+O=A+G+R+O+K11$

And so you can reduce to $K+N=G+R+K11$

Now you have to put the numbers in order to have the maximum and the minimum respecting the boundary conditions.

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...

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  • $\begingroup$ 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, $K=9, N=5, G=1, R=2$. $\endgroup$ – TonyK Feb 12 '16 at 10:33
  • $\begingroup$ Sorry, I did a mistake. Edit my answer if you want or I'll correct later. Oherwise post a new one and I'll delete mine. $\endgroup$ – Tarlo_x Feb 12 '16 at 11:24

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