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

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Modern Puzzles by Henry Ernest Dudeney: $56$

Easy Division
To divide the number $8 \, 101 \, 265 \, 822 \, 784$ by $8$, all we need to do is transfer the $8$ from the beginning to the end!
Can you find a number beginning with $7$ that can be divided by $7$ in the same simple manner?


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:

$N = 7 \, 101 \, 449 \, 275 \, 362 \, 318 \, 840 \, 579$

$\blacksquare$


Sources

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