Henry Ernest Dudeney/Puzzles and Curious Problems/112 - Simple Division/Solution 4

Puzzles and Curious Problems by Henry Ernest Dudeney: $112$

Simple Division

There is a simple division sum.

Can you restore it by substituting a figure for every asterisk, without altering or removing the sevens?

If you start out with the assumption that all the sevens are given, and that you must not use another,

you will attempt an impossibility, though the proof is difficult;

but when you are told that though no additional sevens may be used in divisor, dividend or quotient,

any number of extra sevens may be used in the working.

It is comparatively easy.

             **7**
       -----------
 ****7*)**7*******
        ******
        -------
        *****7*
        *******
        -------
          *7****
          *7****
          -------
          *******
           ***7**
          --------
            ******
            ******
            ------

Solution

$124 \, 974 \times 39 \, 781 = 4 \, 971 \, 590 \, 694$

             39781
       -----------
 124974)4971590694
        374922
        -------
        1222370
        1124766
        -------
          976046
          874818
          -------
          1012289
           999792
          --------
            124974
            124794
            ------

Proof

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