Suppose I have the numbers 18, 6 and i divide them by the gcd 6 to get 3, 1. Is there some way to get back the numbers 18 and 6? Basically is there some way to know what gcd divided the numbers so i can multiply 3, 1 with the same gcd number?
2026-04-01 13:37:27.1775050647
Can I get back original numbers after i divide them by the gcd?
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If I understand you correctly, you're looking at the function $$ f(a,b) = \left(\frac{a}{\gcd(a,b)}, \frac{b}{\gcd(a,b)}\right)$$ and you're asking whether knowing only the pair that results from $f(a,b)$ will allow you to find out what $a$ and $b$ were.
It is easy to see that this is not possible. Your own example is $f(18,6)=(3,1)$, but we also have $f(3,1)=(3,1)$ -- so just knowing that the output of the function was $(3,1)$ will not tell you whether the input was $(18,6)$ or $(3,1)$, or any of the infinitely many other inputs that all yield that particular output.