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Find the Jacob's symbol of

$ \frac{32}{15}$


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$ \frac {32}{15} = \frac {32}{5 \times 3} = \frac {32}{5} \frac {32}{3} ---------1 $ but $ 32 = (5 \times 6) + 2$ $ \therefore ( \frac {32}{5}) = ( \frac {2}{5}) ----------------2 $ $ 32 = (3 \times 10) + 2$ $ \therefore ( \frac {32}{3}) = ( \frac {2}{3}) ----------------3 $ from 1, 2 & 3 $ \frac {32}{15} = \frac {2}{5} \frac {2}{3} -----------------4 $ for odd prime p, ($ \frac {2}{0}) = (-1)^{\frac {P^2-1}{8}}$ When p=3, $ (\frac {2}{3}) = (-1)^{\frac{3^2-1}{8}}= (-1)^1= -1 ----------5$ when p = 5 $ (\frac {2}{5}) = (-1)^{\frac{5^2-1}{8}}= (-1)^3= -1 ------------6$ From 1,2, &3 $ \frac {32}{15} = (-1)(-1)$ $ \frac {32}{15} = 1 $

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