Solution to: Long Division

Replace the dots by letters as follows:

a b 9 / 6 c 8 d e f \ g 5 3
        h i j 2
        -------
          k 9 l m
          n p 4 q
          -------
            r s 4 t
            u v w x
            -------
                  0

Because the long division ends without remainder, we know that:

• uvwx=rs4t
• t=f
• m=e

We conclude:

a b 9 / 6 c 8 d e f \ g 5 3
        h i j 2
        -------
          k 9 l e
          n p 4 q
          -------
            r s 4 f
            r s 4 f
            -------
                  0

In addition, the following holds:

• ab9 × 3 (of g53) always ends on 7, so f = 7
• ab9 × 3 = rs47, so 3 × b + 2 (of 3 × 9 = 27) = 4, so b = 4
• np4q = 5 × a49 and always ends on 5, so q = 5

This results in:

• e = 9
• g × a49 = hij2, so g = 8
• 608097 (smallest possible numerator) divided by 853 gives 712, so a ≥ 7. The largest possible numerator (698997) divided by 853 gives 819. So, the only possibility for a is 7.

Now, the other digits can automatically be derived, since both numerator and denominator are known. The complete long division looks like:

7 4 9 / 6 3 8 8 9 7 \ 8 5 3
        5 9 9 2
        -------
          3 9 6 9
          3 7 4 5
          -------
            2 2 4 7
            2 2 4 7
            -------
                  0

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