BRSAGAR: Rhythm in 1/7

Rhythm in 1/7

The reminders and answer to 1/7 can be written mentally in seconds. Let us see how?

If you divide 1/7, first remainder after decimal point is 3 and next 2.  In first method to find all remainders, start with 3.

  1. i)Multiply 3 by itself to get 9. Take off 7, so the next remainder is 2 (9-7).
  2. ii)Multiply 2 by first remainder 3 to get 6, which is the third remainder.

iii)                 Multiply 6 by first remainder 3 to get 18. Divide by 7 and next remainder is 4.

  1. iv)Multiply 4 by first remainder 3 to get 12. Take away 7 and next remainder is 5.
  2. v)Multiply 5 by first remainder 3 to get 15. Taking away 7s, next remainder is 1.
  3. vi)1 x 3 = 3, the first remainder. So stop because of repetition.

Thus the remainders are 3, 2, 6, 4, 5, and 1.

Second method: Multiply any remainder by the first and subsequent remainders to get next remainders as multiply 6 by 3 to get 4 as next remainder, then 6 by 2 to get fifth and 6 by 6 to get the sixth remainder. Similarly 4 by 3 gives fifth remainder and 4 by 2 the sixth remainder.

Rhythm: In 3, 2, 6, 4, 5, 1, 6 is 7 – 1 (denominator – numerator of 1/7). This means up to 6, half the job is done and calculate other remainders as compliments of 7 as 7 – 3 =4, 7 – 2 = 5 and 7 – 6 = 1. See, the symmetry of 3, 2, 6

4, 5, 1

Here, sum of each column is 7 as 3+4, 2+5 and 6+1.

Now the answers. Multiply each remainder by 7 and take down only unit digits.

Multiplying 3, 2, 6, 4, 5, 1 by 7, you get 21, 14, 42, 28, 35, and 7. Taking only their unit digits, the answer of 1/7 is .142857. Here also, there is symmetry, the last three are compliments of 9 of the first three as 9-1=8, 9-4=5, and 9-2=7.

 

BRSAGAR

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