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Old 2021-09-09, 02:09   #36
Dr Sardonicus
 
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Feb 2017
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Quote:
Originally Posted by MattcAnderson View Post
Hi again all,

My new favorite seven digit number is 9,999,999.
<snip>
9,999,999 = 3*3*239*4649.
<snip>
This brought to mind the following example illustrating the properties of the blocks of digits in the repeating decimals for k/p, k = 1 to p, p prime. Here, p = 239, a factor of 9999999. Since 239 divides 9999999, the repeating decimal for k/239, k = 1 to 238, has period 7. The blocks of digits may be further classified by the following 34 numbers:

[1004184, 2887029, 2050209, 1464435, 1046025, 1757322, 1255230, 1087866, 1506276, 3054393, 1380753, 1966527, 3765690, 2635983, 1882845, 1631799, 1422594, 1589958, 1129707, 4979079, 3556485, 3974895, 2426778, 4476987, 1213389, 2384937, 4560669, 1924686, 2803347, 2343096, 1673640, 1171548, 2008368, 4058577]

Each of these 34 numbers is the 7-digit block for the repeating decimal for a fraction k/239, for some k between 1 and 238. In each case, the value of k is obtained by dividing the 7-digit number by 41841. For example, dividing 1004184 by 41841 gives 24, and 24/239 = .10041841004184...

By cyclically permuting the digits in each block, the seven-digit blocks of the repeating decimals for the remaining fractions k/239, k = 1 to 238, are obtained.

For example, cyclically permuting the digits of the first block gives 0041841, and .00418410041841... = 1/239; permuting again, 10/239 = .04184100418410..., 100/239 = .41841004184100..., 44/239 = .18410041841004... and so on.

Each of the numbers is the smallest cyclic permutation of the block of digits whose leading digit is non-zero (i.e, a number between 1000000 and 9999999).

Last fiddled with by Dr Sardonicus on 2021-09-09 at 02:16 Reason: elaborate on permuting blocks of digits
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