The twins Maxim & Minerva independently replaced the letters in the word
KANGAROO upon the following rules:
1. Similar letters are replaced by the same digit
2. The resulting 8-digit number must be divisible by 11.
3. Maxim had to choose the maximal qualifying number, Minerva - the minimal.
Comparing their correct answers they've found out that a certain letter in the word KANGAROO was replaced by the same digit in their independent answers.
Specify WHAT letter and explain WHY.
I'll assume the basics: no leading zeros and each letter represents a distinct base 10 digit.
For KANGAROO to be a multiple of 11, the difference (K+N+A+O)-(A+G+R+O) must also be a multiple of 11. This difference simplifies to K+N-(G+R) = 0 mod 11.
Let Minerva's first three digits be the absolute minimum with the assumptions previously mentioned. Then KAN=102. This implies 3-(G+R) = 0 mod 11 with G and R limited to {3,4,5,6,7,8,9}. The smallest G can be in this equation, given the set of digits available, is G=5, which makes R=9. Then Minerva has KANGAROO=10250933.
Similarily, Maxim's first three digits are KAN=987. This implies 16-(G+R) = 0 mod 11 with G and R limited to {0,1,2,3,4,5,6}. Then the largest G is 5 with R=0. Then Maxim has KANGAROO=98758066.
Then comparing the numbers is simple:
KANGAROO
10250933
98758066
The letter/digit they have in common is G=5.
Solution
