## 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
``` Back to the puzzle
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