Using identities from Euclid's algorithm to solve problems I have been given the following problems
(a) Use the $GCD$ algorithm to compute the greatest common divisor of $546$ and $416$. 
(b) Use your working for the previous part to express $\frac{416}{-546}$ as a rational ${m \over n}$ where $n\geq 1$ and $\gcd(m,n)=1$.
I am able to do part  (a) and was left with the following working
$\gcd(416,546)$
$\rightarrow 546 = 1 * 416 + 130$
$\gcd(130,416)$
$\rightarrow 416 = 130 * 3 + 26$
$\gcd(26,130)$
$\rightarrow 130 = 5 * 26 + 0$
$\gcd(0, 26)$
therefore $\gcd(416,546)=26$
Then with part (a), I found that because
$416 = 16 * 26$ and $546 = 21 * 26$
That ${16 \over -21}$ is the answer, this is wrong, and I'm struggling to see what I need to do to begin on this problem

If $gcd(m,n)=1$ then $m/n$ is a fraction reduced to its lowest terms, so you've actually used that fact. 
And as I said in the comments, the other condition in part b) is that you have to ensure that $n≥1$, which is why your original answer was wrong.  
