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

I just wanted to point out that, given two integers $a,b\ge 1$,the Extended Euclidean Algorithm, continued one step further, gives coprime coefficients $u,v$ such that $ua=vb=0$, hence both gives $ \operatorname{lcm}(a,b)=\lvert u\rvert\, a=\lvert v\rvert\,b$, and the expression of $\dfrac ab$ or $\dfrac ba$ in lowest terms.
Here:
$$
    \begin{array}{r@{\qquad}r@{\qquad}r@{\qquad}r}  
r_i & u_i & v_i & q_i\\
   \hline
 546 & 1 & 0 & \\
  416 & 0 & 1 & 1\\
     \hline
 130 & 1& -1 & 3\\
 26 & -3 & 4 & 5 \\
0 & 16 & -21 & \\
     \hline
    \end{array} $$
We deduce at once $\;\gcd(546,416)=26$,  $\;\operatorname{lcm}(546,416)=21\cdot416=8736$ and 
$$\frac{416}{-546}=-\frac{16}{21}.$$
