Shesanyankena Charamena- "The remainders by the last digit"

Consider: 1/7. When divising 1(0) by 7 the remainder is 3. Therefore, dividing 3 by 7 will subsequently lead to remainder 9 (= 3x3). But since 9 is more than 7 the remainder would be 2, so the remainder sequence is:

         3, 2

Now 2 divided by 7 will have remainder of 6 (3x2), that is

3, 2, 6

Continuing

3, 2, 6, 4, 5, 1

We stop when the remainder sequence starts to repeat. Now, multiply these remainders by the last digit (7) of the denominator and keep only the first digit (LSD). So we have:

7x3 = 21 => put down 1

.1               

 3, 2, 6, 4, 5, 1

7x2 = 14 => put down 4

.1  4           

 3, 2, 6, 4, 5, 1

7x6 = 42 => put down 2

.1  4  2       

 3, 2, 6, 4, 5, 1

Continuing

.1  4  2  8  5  7         

 3, 2, 6, 4, 5, 1

So the answer is 1/7 = .142857142857..........