Question

You know how 1/7=0.142857.Can u predict what the decimal expansions of 2/7,3/7,4/7,5/7,6/7 are without actually doing long division? If so, how?

Answer 1

1/7 = 0.142857
2/7 = 0.285714
3/7 = 0.428571
4/7 = 0.571428
5/7 = 0.714285
6/7 = 0.857142 
7 is a very special number, with 1/7 is a rational with (unterminatng) 
decimal of period 6. In fact, 1/7 = 0.[0.142857] (cyclic of period 6)
Also, other fractions i/(2 <=i <= 6) are also cyclic in these 6 numbers
without changing the order, as you can see above) 
In group theory, they froma cyclic group of order 6, namely
Z7*,*) ,where Z7* = {[0],[1],[2],[3],[4],[5],[6]} 
I am not quite sure about "Clock diagram"
One way you can try is to draw a clockwise rotation (regular hexagon with
vetices marked in the order as) 

142857 --> 428571 --> 285714 --> 857142 -->571428 --> 714285
1/7 --> 3/7 --> 2/7 --> 6/7 --> 4/7 --> 5/7 
Like 1/7
*3 / \*3
5/7 / \ 3/7 
| |
*3| |*3
4/7 2/7
\ /
*3 \ /*3
6/7 
[Sorry for hard to have good diagram here.] 
Note 3/7 = (10*1/7 -1), 2/7 =(10 * 3/7 -4) , 6/7 =(10* 2/7 -2) ,
4/7 =(10* 6/7 -8) , 5/7 =(10* 4/7 -5)
If you know modulus(remainder dividing by 7)
, you can see 1/7 * 3 = 3/7, 3/7*3 = 9/7 = 2/7,
2/7 * 3 = 6/7, 6/7 * 3 = 18/7 = 4/7, 4/7 * 3= 12/7 = 5/7 and 5/7*3 = 15/7 = 1/7.
(this means each fraction is 3 times ofthe previous one.
I.E. (i+1) mod 7/ 7 = (3* i) mod 7/7 for all 1<=i<= 5. 
Good luck !! HOPE IT HELPED..

Answer 2

Answer:

Mark as brain list. ......