Question

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In the expression below, each letter represents a one-digit number. Where the same letter appears, it represents the same number in each case. Each distinct letter represents a different number. In order to make the equation true, what number must replace C?
AAA
+AAB
ABC
--------
2012

Answer 1

@amistre64 @ranga @radar @robtobey @AkashdeepDeb @phi @.Sam.

Answer 2

@skullpatrol

Answer 3

AAA = 100A + 10A + A = 111A
AAB = 100A + 10A + B = 110A + B
ABC = 100A + 10B + C

Add them all up: 111A + (110A + B) + (100A + 10B + C) =

321A + 11B + C = 2012 ---- (1)

What is the maximum allowed for A?
2012 / 321 = 6.27. So the maximum A can be is 6. If A were 7, 321*7 = 2247 which exceeds 2012 on the RHS of (1). So max for A is 6.

Can A be anything less than 6? What if A were 5? Put A = 5 in (1):
321*5 + 11B + C = 2012 or 11B + C = 407.
The maximum 11B + C can be is 99 + 9 = 108.
So 11B + C cannot be 407.
Therefore A cannot be anything less than 6.

Put A = 6 in (1)
321*6 + 11B + C = 2012

11B + C = 86 --- (2)

86/11 = 7.8. So the maximum B can be is 7. If B falls short even by 1, the total will come down by 11 which a single digit C alone cannot compensate to add to 86 in (2). So B must be 7.

A = 6, B = 7
11*7 + C = 86
C = 86 - 77 = 9

A = 6, B = 7, C = 9