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The following table represents addition of two six-digit numbers given in the first and the second rows, while sum is given in the third row. In the representation, each of the digits $0,1,2,3,4,5,6,7,8,9$ has been coded with one letter among A,B,C,D,E,F,G,H,I,J,K, with distinct letters representing distinct digits.

      B  H  A   A   G   F
  +     A  H  J   F   K   F
    A   A  F  G   C   A   F

Which digit does the letter A represent _______

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 A=1

I’ll try to reason out how I decoded all the encodings in this question and will paste the reference link in the subsequent questions.

  B(9) H(5) A(1) A(1) G F(0)
+ A(1) H(5) J F(0) K F(0)
A(1) A(1) F(0) G C(2) A(1) F(0)
  1. A good place to start is to figure out A & F. It’s given that F+F=F, this is only possible when F=0. A is an overflow digit of 5 digit addition for 2 numbers, this implies that A can only be 1. Max possible result of 2 digit addition including Carry-In from the previous digit is 19 and 1 is the overflow digit.
  2. B is the second target for decoding, B+1=1, this is only possible if 9+1=10 and another 1 carried in from previous digit addition is included making it 9+1+1=11. So, B=9.
  3. H+H=F=0, either H is 0(which isn’t possible as 0 has been assigned to F and every encoding is distinct). So, the only possibility is H=5 & 5+5=10 where F=0 and carry is 1.
  4. A+F=C or 1+0=C, either C can be 1 or 2(if a carry of 1 is present from the previous digit). 1 isn’t permissible for C as it’s taken by A. Therefore, C=2.
  5. G=J+1 so G can be 4,7,8 and correspondingly J can be 3,6,7.
  6. As G+K has a digit 1 and the carry in=0, G+K=11, now this can be achieved in 4+7,7+4,8+3 or K can be 7,4,3.
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