1. ## [SOLVED] Z18 Calculation

$\displaystyle \frac{2}{7}+\frac{7}{13}=\frac{[2]X[13]+[7]X[7]}{[7]X[13]}$

Im not sure how to multiply this out please show steps on how to do this. Thanks

2. Originally Posted by ronaldo_07

$\displaystyle \frac{2}{7}+\frac{7}{13}=\frac{[2]X[13]+[7]X[7]}{[7]X[13]}$

Im not sure how to multiply this out please show steps on how to do this. Thanks
From the Euclidean algorithm (or trial and error)

$\displaystyle 7^{-1}=13 \mod(18)$ and

$\displaystyle 13^{-1}=7 \mod(18)$

so we end up with
$\displaystyle 13 \cdot 2 +7\cdot 7 =26+49=75 \equiv 3 \mod(18)$

3. How did you get from 75 to 3mod(17)?

4. Hi as you mentioned it's not 3 mod 17, actually it is 3 mod 18.

take 75 and when you try to divide 75 by 18, you will get 3 as a remainder and hence 3 mod 18. is it helpful...?

5. Here is another way. To find inverse of 7 you need to solve $\displaystyle 7x\equiv 1(\bmod 18)$.
This is equivalent to $\displaystyle 7x\equiv 1 - 2\cdot 18(\bmod 18)\implies 7x\equiv -35(\bmod 18) \implies$$\displaystyle x\equiv -5(\bmod 18)\implies x\equiv 13(\bmod 18)$.