The present paper analyses and presents several improvements to the algorithm for finding the $(a,b)$-pairs of integers used in the $k$-ary reduction of the right-shift $k$-ary integer GCD algorithm. While the worst-case complexity of Weber's ''Accelerated integer GCD algorithm'' is $\cO\l(\log_\phi(k)^2\r)$, we show that the worst-case number of iterations of the while loop is exactly $\tfrac 12 \l\lfloor \log_{\phi}(k)\r\rfloor$, where $\phi := \tfrac 12 \l(1+\sqrt{5}\r)$.\par We suggest improvements on the average complexity of the latter algorithm and also present two new faster residual algorithms: the sequential and the parallel one. A lower bound on the probability of avoiding the while loop in our parallel residual algorithm is also given.