I was reading a certain book in which certain divisibility rules for numbers were given, in it was a given that if any number is divided by 7 & if the fraction were recurring then the number of digits that would recur at maximum would be only six, I am unable to understand the reason behind this.
I do not know the reason why, but 1/7 = 0.142857 repeating. Then any multiple of that gives the same recurrence, but may start with a different place. For example, 3/7 = 0.42857142857, and the 142857 repeats. Notice it's the same recurrence, but it begins at a different place this time. Try this: for all integers n, with, find n/7. You should find that you get the same 6 digits repeating in the same order, but starting at a different value. Once you get to n=7, then 7/7=1. Beyond that, for n=8, 8/7= 1+1/7, so you just repeat the same pattern, but with a 1 in front of the decimal place. Now, I do not know why the decimal repeats for every integer divided by 7, but maybe you can figure it out by showing what happens when you add the digits of two multiples of 1/7?
Also, notice that there are 6 integers between 0 and 7, and for each, the decimals begin at at a different number, but repeat in the same order. If for two integers less than 7, n/7 and m/7 began at the same number and repeated in the same pattern, then n/7 would be equal to m/7 where n,m<7. That is impossible - it's like saying 1/7=4/7.
Of course this still doesn't answer why those 6 repeat in the first place, or why any integer divided by 7 repeats those same 6 in the same order.
I know nothing about number theory, so I may not have been able to help at all. I just thought it would be fun to examine this and think about it with you! Good luck, I hope I helped somewhat..
"Pure mathematics is, in its way, the poetry of logical ideas."