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How do you convert

$14\frac{8}{13}$

into base 3?

I was able to convert $\frac{3}{7}$ into base 3 by constantly multiplying by 3 and dividing the numerator by denominator until I finally got a repetition, but this method doesn't seem to work for $14\frac{8}{13}$.

The correct answer is supposed to be $112.\overline{121}_{3}$

  • 0
    Oops, yes, I meant added.2011-11-05

2 Answers 2

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Just like you convert a fraction to decimal. $8\cdot 3=24=13(1)+11,$ so the first digit (ternit?) is $1. \ \ 11\cdot 3=33=13(2)+7,$ so the second digit is $2. \ \ 3\cdot 7=21=13(1)+8$ and so on. I get $112.\overline{121}_{3}$

Alternately you can notice that $\frac{8}{13}=\frac{16}{3^3-1}$, so the repeat is $3$ digits long and is $16_{10}=121_3$

  • 0
    Typo, I meant to write $\frac{8}{13}$. Thanks!2011-11-05
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You convert 8 and 13 into base 3, and do a long division in that base,

          0.202           8 = 101       ------------           -----   111 ) 101              13 = 111          22.2          -----           1.100           0.222             101  (repeats on three places) 

So the answer is 112.202 202 202