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186X35 is the second-smallest cyclic number, corresponding to the prime 7 (the smallest cyclic number is 2497, corresponding to the prime 5). Note that {186X35, 2497} uses all the digits (except the digits that is not coprime to 10-1 (=E), i.e. {0, E}) exactly once.

186X35, the six repeating digits of 1/7, 0.186X35, is one of the two best-known cyclic numbers in base 10 (the other is 2497, the four repeating digits of 1/5, 0.2497). If it is multiplied by 2, 3, 4, 5, or 6, the answer will be a cyclic permutation of itself, and will correspond to the repeating digits of 2/7, 3/7, 4/7, 5/7, or 6/7 respectively.

Unlike the similar number in base X: 142857X, which is both Kaprekar number and Harshad number in base X, 186X35 is neither Kaprekar number nor Harshad number in base 10.

Calculation Edit

1 × 186,X35 = 186,X35
2 × 186,X35 = 351,86X
3 × 186,X35 = 518,6X3
4 × 186,X35 = 6X3,518
5 × 186,X35 = 86X,351
6 × 186,X35 = X35,186
7 × 186,X35 = EEE,EEE

If multiplying by an integer greater than 7, there is a simple process to get to a cyclic permutation of 186X35. By adding the rightmost six digits (ones through hundred thousands) to the remaining digits and repeating this process until only six digits are left, it will result in a cyclic permutation of 186X35:

186X35 × 8 = 1186X34
1 + 186X34 = 186X35
186X35 × 497 = 8286X28E
82 + 86X28E = 86X351
186X352 = 2E3224X9381
2E322 + 4X9381 = 5186X3

Multiplying by a multiple of 7 will result in EEEEEE through this process:

186X35 × 259 = 42EEEE79
42 + EEEE79 = EEEEEE

The result cannot contain either any digit of 1/5 (2, 4, 9, 7) or the zero digit (0).

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