Question 1204315
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The three digit number ABC means 
A = hundreds digit
B = tens digit
C = units or ones digit


ABC is more formally written as 100A+10B+C
AC becomes 10A+C


Divide those values and we get a quotient of 11 and no remainder.


(100A+10B+C)/(10A+C) = 11
100A+10B+C = 11(10A+C)
100A+10B+C = (10+1)(10A+C)
100A+10B+C = 10(10A+C)+1(10A+C)
100A+10B+C = 100A+10C+10A+C
10B+C = 10C+10A+C
10B = 10C+10A
0 = 10A-10B+10C
10(A-B+C) = 0
A-B+C = 0
A = B-C


To make number ABC as large as possible, we need A as large as possible.
At the same time, we need B to be as large as possible as well.
For unique single digits B and C, B-C maxes out when these digits are as far away from each other as possible, and when B > C.
That happens when B = 9 and C = 0
So A = B-C = 9-0 = 9


But A = 9 and B = 9 overlap.
We assume that A and B are different values. Otherwise the number ABC would be AAC or BBC.
Let's go for B = 9 and C = 1 instead.
A = B-C = 9-1 = 8
This would allow A,B,C to be different integers.


The number ABC = 891 is the largest possible value so that ABC/AC = 11 without a remainder (i.e. remainder is 0).



More specifically, 
891/81 = 11



Answer: 891
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