Problem 358

Cyclic numbers

A **cyclic number** with `n` digits has a very interesting property:

When it is multiplied by 1, 2, 3, 4, ... `n`, all the products have exactly the same digits, in the same order, but rotated in a circular fashion!

The smallest cyclic number is the 6-digit number 142857 :

142857 1 = 142857

142857 2 = 285714

142857 3 = 428571

142857 4 = 571428

142857 5 = 714285

142857 6 = 857142

The next cyclic number is 0588235294117647 with 16 digits :

0588235294117647 1 = 0588235294117647

0588235294117647 2 = 1176470588235294

0588235294117647 3 = 1764705882352941

...

0588235294117647 16 = 9411764705882352

Note that for cyclic numbers, leading zeros are important.

There is only one cyclic number for which, the eleven leftmost digits are 00000000137 and the five rightmost digits are 56789 (i.e., it has the form 00000000137...56789 with an unknown number of digits in the middle). Find the sum of all its digits.

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