Problem 1068

It cannot be done. Assume it can be done. Now consider an integer point $A$, assume it is labeled with a positive integer $a$. Let $a$ have $n$ distinct prime factors.

Take another integer point $A_{1}$ in the plane. Clearly, there are other integer points $B_{1}$ on the line $A A_{1}$, for example, $B_{1}$ can be taken as the symmetric point of $A$ with respect to $A_{1}$.

Since the 3 numbers written on $A, A_{1}, B_{1}$ have a common divisor greater than 1, they can all be divided by some prime $p_{1}$. In particular, $p_{1} \mid a$.
Take another integer point $A_{2}$ in the plane, such that $A_{2}$ is not on the line $A A_{1}$.
There are other integer points $B_{2}$ on the line $A A_{2}$, the 3 numbers written on $A, A_{2}, B_{2}$ can all be divided by some prime $p_{2}$. In particular, $p_{2} \mid a$.
Since $A, A_{1}, A_{2}$ are not collinear, $p_{1} \neq p_{2}$. Continue this process to construct lines $A A_{3}, A A_{4}, \cdots, A A_{n+1}$, each time obtaining a new prime that can divide $a$, resulting in a total of $n+1$ distinct primes, all of which can divide $a$. This contradicts the assumption that $a$ has only $n$ distinct prime factors.