Auxiliary fractions observation
 

I was watching a vedic math YouTube video on the concept of auxiliary fractions and noticed something about the prepended remainders. I am not well educated in mathematics and lack the proper terminology, so I have attached a photo. I'll do my best to explain an example and welcome any insight. Most likely a well known concept, but I'd like to learn more!

Ex: 1/39 = Auxiliary fraction first order 0.1/4
0.[SUB]1[/SUB]0[SUB]2[/SUB]2[SUB]2[/SUB]5[SUB]1[/SUB]6[SUB]0[/SUB]4[SUB]0[/SUB]1
The prepended remainders 001221[SUB]4[/SUB]= 105[SUB]10[/SUB]
AND
((40^6)-1)/399 = 105,025,641
*notice the 105 prepended to the repetend of 1/39.
(Also noted that 4095[SUB]10[/SUB] = 333333[SUB]4[/SUB]

I've worked out dozens of examples in many other bases and the same is always true. Another example is included in my photo.

Looks remarkably similar to the discussion in [url=https://mersenneforum.org/showthread.php?t=24331]this thread[/url].