## Solution to: Long Division

Replace the dots by letters as follows:

a b9 / 6c8d e f\g5 3h i j2------- k9l m n p4q ------- r s4t u v w x ------- 0

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

- uvwx=rs4t
- t=f
- m=e

We conclude:

a b9 / 6c8d e f\g5 3h i j2------- k9l e n p4q ------- r s4f r s4f ------- 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 =
7) = 4, so b = 4*2* - 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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