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

by : $56$

 * Easy Division

Proof
Let $N$ 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:

0.10144923 69)7.00000000000  6.9     130   ---      69      100    ​  69 ​    ---      310      276      ---       340       276       ---        640        621        ---         190         138         ---          220          207          ---           130

{{wip|}