Find pairs $(a,b)$ with $\gcd(a,b),\gcd(a + 1, b),\ldots, \gcd(a + k, b)$ given

Question

Given a set of GCD's, how to find a set of numbers that satisfy all their criteria?
Suppose we are given a $k$ integers $\gcd(a,b),\gcd(a + 1, b),\ldots, \gcd(a + k, b)$ for some k.
How to get a and b from this? Given the GCD's, I need to find (a,b).
Example: if 4 GCD's are given 3,2,1,6, then the pair (a,b) which satisfy the above condition is (3,6).
Any help will be useful!

Answer 1

Hints: Let $g_0,g_1,\ldots,g_k$ be the specified GCDs. Clearly one must have $\text{LCM}(g_0,\ldots,g_k) \mid b$. Show that if $(a,b)$ is any solution then we may replace $b$ by $b_0 := \text{LCM}(g_0,\ldots,g_k)$, so that $(a,b_0)$ is also a solution.

Therefore we are free to assume $b = b_0$. For each prime power factor $p^r$ of $b_0$, there must be at least one $g_i$ that is divisible by $p^r$. This uniquely determines $a$ mod $p^r$. Now Chinese Remaindering determines $a$ mod $b_0$.

There is at most one working choice of $a$ mod $b_0$. Prove that if any one value of $a$ satisfies the full set of constraints, then so does every $a$ in that congruence class. This provides all solutions of the specific form $(a,b_0)$; there are other solutions with larger values of $b$, and these are more complicated to describe — but you only are looking for one.