i dare say many many of you will know that when you divide one number by another you get a repeating sequence [* qualified ]

example 1/7 = .142857 142857 142857 ..

now suppose instead of conventional division we divide in excess of the number each time [overdivide]

so 10/7 = 2 [ie 14] and the remainder is now 4

40/7 = 6 and the remainder is now 2

20/7 = 3 and the remainder is now 1 so were back where we started but with a shorter sequence .. this being 263 as opposed to 142857

and instead of a sequence which approaches [but never reaches the true value ???] we now have an Oscillating approximation that bounds the value

by this i mean .. you go up 2 then back 6 [ = 14] then up 3 [=143] then back 2 [=1428] then up 6 [=14286] then back 3 [=142857] etc

There are a whole series of rules that can be deduced about the lengths of sequences

[in the case of 1/prime these sequences have to divide the prime = prime -1 [eg 7 is 6 long ...13 happens also to be 6 long ..(13-1)/2 ... 41 is 5 long and so is 271 ...

239 and 4649 have lengths of 7 etc etc ]

and the results are evident in the following factors of 9 recuring-length-X [ie X-1 recurring which is what is effectively happening in division length X and which may then be reduced to finding the factors of (3x3x ) ...1 recuring-length-X .........]

obviously this enables the prediction of division sequences BUT only if you have the prime components of both the 'thing being divided' and the divisor

i havent got access to all my stuff at the moment but

i hope you find this interesting

regards mike b

here this is described as recreational mathematics .. the idiot who wrote that didnt understand the process of division as i suspect most dont

https://ipfs.io/ipfs/QmXoypizjW3WknFiJnKLwHCnL72vedxjQkDDP1mXWo6uco/wiki/Repunit.html