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\[gcd(a,b) = gcd(amodb,b)\]  Let $x = amodb$. amodb$, and let k be any number.  \[x-a = bk\] for some k. \[x=a+bk\]  Let $k=-1$     \[x=a-b\]     \[gcd(a,b) = gcd(a-b,b)\]     If we subsitute $5x+3y$ for $a$, and $3x+2y$ for $b$, it will simplify to $(x,y)$   \item   Using the above reasoning for finding GCD, we ahve to pick a lever $k$ that is an integer and will make $a$ reduce to 1 in $(a,b)$.     \[(S_{n} - kS_{x}, S_{x})\]  \end{itemize}