I have measured a quantized (or discrete) physical quantity. This means that there exists a fundamental quantity $e$ such that the physical quantity is an integer $n$ times the fundamental quantity $e$. For example, if I measure $1,3,2,4$ then $e$ is $1$. The quantities measured are fixed positive real numbers and they will be of the form $n_1e, n_2e,\ldots,n_ke$. What is an efficient algorithm for finding $e$? This is equivalent to finding the set of positive integer numbers $n_1,\ldots,n_k$.
I appreciate your help.
Most likely, $$e=\gcd(n_1e,\dots,n_ke)$$ where $\gcd$ is the greatest common divisor function. Actually, the right hand side is $$\gcd(n_1e,\dots,n_ke)=e\cdot\gcd(n_1,\dots,n_k),$$ where $\gcd(n_1,\dots,n_k)$ is likely equal to one. This function can be calculated recursively for $k>2$. If $$g_i=\gcd(n_1,\dots,n_i)$$ for $1\le i\le k$, then $g_{i+1}=\gcd(g_i,n_{i+1})$. GCDs of two arguments can be found using the Euclidean Algorithm, which successively replaces one of the two arguments, discarding their signs, by its remainder modulo the other, terminating with the last nonzero remainder.
Also, it is interesting to note that the probability of two random integers being coprime (having $\gcd=1$) is $\frac6{\pi^2}$, so the probability of $k$ random integers all being pairwise coprime would be $$1-\left(1-\frac6{\pi^2}\right)^k.$$ But this probably won't apply to your observations unless the $n_i$ are all far from unity. For $n_i$ close to $1$, one would need to know their distribution to estimate this likelihood.