Given a number '2/7'.Find the 57th number after the decimal point.

Whats the method?

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- Aug 3rd 2010, 10:36 AMsfspitfire23No idea where to start
**Given a number '2/7'.Find the 57th number after the decimal point.**

Whats the method?

- Aug 3rd 2010, 10:42 AMundefined
Look at the repetend length (period). Well it's 6. So reduce 57 mod 6 and conclude that the 57th digit after the decimal point is the same as the 3rd digit after the decimal point, which is 5.

The 1st digit after the dec point is the same as the 7th and the 13th and the 19th... so you can get there just with pattern recognition. - Aug 3rd 2010, 11:06 AMArchie Meade
How to notice the pattern is as follows...

divide 2 by 7 and observe the carryover

$\displaystyle 7)2.00000000000$

As the division is performed, find the first time that the carry repeats itself.

7 into 2...doesn't divide

7 into 20 is 2 rem 6.

First carry is 6, so the next division is 7 into 60

7 into 60 is 8 rem 4

7 into 40 is 5 rem 5

7 into 50 is 7 rem 1

7 into 10 is 1 rem 3

7 into 30 is 4 rem 2

7 into 20.... there is where the pattern repeats since we always have a zero after the remainder

for the next division,

hence the division yields 0.285714**285714**285714**285714**......

Then Undefined's logic has covered the rest.