Suppose I have the following equation $$\frac{ax+by}{t}= \text{an integer}$$ and $a$, $b$ and $t$ are known. I also have a set of solutions for $x$ and $y$ out of which only one will satisfy the above equation. Without having to resort to trying each solution out, which is the fastest way (if it exists) to figure out the correct values of $x$ and $y$? Does something like a Chinese remainder theorem help here? And how to proceed if $a$ and $b$ are co-prime?
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See:https://math.stackexchange.com/questions/20906/how-to-find-an-integer-solution-for-general-diophantine-equation-ax-by-cz?rq=1 – NoChance May 13 '19 at 22:22
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Thanks but that does not help much. What if $a$ and $b$ are co-prime? Then their gcd is 1 and I back to square one. – RTn May 13 '19 at 23:05
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are $a,b,t$ real , integers? – G Cab May 13 '19 at 23:12
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Yes, indeed they are integers and real! – RTn May 13 '19 at 23:13
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What is known about gcd(a,t) ? gcd(b,t) ? – May 14 '19 at 15:02
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gcd(a,t) and gcd(b,t) both are co-prime. – RTn May 14 '19 at 20:09