So we are given the expression, $$ (a-b)/2*s + b*t = d, $$ where $a,b,s$ are odd integers and $t$ is even an integer. Is there some way to rewrite this in the form $$ax + by = d?$$ Where $x$ and $y$ are integer values. This question is relevant because if one tries to construct the recursive binary extended Euclidean algorithm, one has to keep track of the Bezout coefficients. For one of the steps, this computation is required, and I'm quite lost as to how to reach the conclusion. I tried to reduce each of the integers with respect to their parity but that made a huge mess.




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