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\item  For this to be true, $x$ and $y$ must be relatively prime. From this, we can observe a pattern that forms where there are unique pairs for (x,y) until the $x*yth$ pair (or index). In our example, $x=5$, $y=7$, and it will have unique pairs until the pattern repeats on the $35th$ index (which is x*y).  \item  I dont know. Since x and y are prime the $gcd(x,y) = 1$.     Using Bezout's identity we know $y(y^{-1}modx + x(x^{-1}mody)=1$  \item  In the case where there are three primes $x,y,z$, the properties from above still hold. There will be unique pairs until the $x*y*zth$ index.  \end{itemize}