Henry Ernest Dudeney/Modern Puzzles/56 - Easy Division/Solution

by : $56$

 * Easy Division

Solution

 * $7 \, 101 \, 449 \, 275 \, 362 \, 318 \, 840 \, 579$

Proof
Let $7 N$ be the number in question.

This is an example of:
 * Integer which is Multiplied by Last Digit when moving Last Digit to First

except here we are moving the first digit to last and dividing by it.

Hence $N$ is the recurring part of the fraction:


 * $q = \dfrac {a_1} {10 a_1 - 1}$

where in this case $a_1 = 7$, giving us:
 * $g = \dfrac 7 {69}$

We perform long division and see what we get:

0.101449275362318840579710... --- 69)7.000000000000000000000000000  6.9     345    552   ---     ---    ---     100    250    280    ​  69    207    276 ​    ---    ---    ---      310    430     400      276    414     345      ---    ---     ---       340    160     550       276    138     483       ---    ---     ---        640    220     670        621    207     621        ---    ---     ---         190    130     490         138     69     483         ---    ---     ---          520    610      70          483    552      69          ---    ---      --           370    580      100           345    552      ...

and we see that:
 * $N = 7 \, 101 \, 449 \, 275 \, 362 \, 318 \, 840 \, 579$