# Thread: fractional number in mode 9

1. ## fractional number in mode 9

what is (1/2)^1000 in mode 9?

I tried dividing 2 by 9, then 4 by 9, then 8 by 9 etc.... and get the remainders in each one....
(although these are decimals but i assume it is ok?) and after 2^-7 it repeats, so it means every 6 exponent is one group, so i divided 1000 by 6 and it gives remainder of 4, and then read what 2^-4 was, which was 7 but the answer says it should be 4... what am i missing?

2. ## Re: fractional number in mode 9

$\displaystyle \frac{1}{2}=2^{-1}\equiv 5 \pmod 9$

$\displaystyle \left(\frac{1}{2}\right)^{1000}\equiv 5^{1000}\equiv 4 \pmod 9$

3. ## Re: fractional number in mode 9

Originally Posted by Idea
$\displaystyle \frac{1}{2}=2^{-1}\equiv 5 \pmod 9$

$\displaystyle \left(\frac{1}{2}\right)^{1000}\equiv 5^{1000}\equiv 4 \pmod 9$
sorry, how is 2^-1 =5 in mode 9?

4. ## Re: fractional number in mode 9

First, in English at least, it is "mod" (short for "modulo"), not "mode"

To answer your question, 1/2 is the multiplicative inverse of 2. That is 2(1/2)= 1. And 2(5)= 10= 9+ 1= 1 (mod 9).