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.

循环数

n位循环数是指拥有如下有趣性质的数：
当它乘以1、2、3、4、……、n时，所有的乘积拥有按相同顺序排列的相同数字，只是在循环地轮转而已！

最小的循环数是6位数142857：
142857 × 1 = 142857
142857 × 2 = 285714
142857 × 3 = 428571
142857 × 4 = 571428
142857 × 5 = 714285
142857 × 6 = 857142

下一个循环数是16位数0588235294117647：
0588235294117647 × 1 = 0588235294117647
0588235294117647 × 2 = 1176470588235294
0588235294117647 × 3 = 1764705882352941
……
0588235294117647 × 16 = 9411764705882352

注意对于循环数而言，前导零是很重要的。

满足以下条件的循环数是唯一的：其最左侧的11位数字为00000000137，最右侧的5位数字是56789（也就是说，它的形式是00000000137……56789，中间的数字未知）。求它的所有各位数字之和。