About two sets of natural numbers

I would like to know if you know by which number should I multiply a list of natural numbers so that it never overlaps with a list of another set of natural numbers at some point in the future. The issue is that we have two different datasets with IDs that are natural numbers and we cannot modify any of the IDs unless we multiply them by a specific number, and we don't want that at some point in time the ID of one dataset finds another similar ID in the other dataset in the future. My limited math knowledge tells me that when taken to the infinite, there will be a clash of IDs at some point just because naturally there's no way to avoid the IDs in one dataset to happen at some point in the other dataset. But maybe I'm wrong, could you please help me?

Thank you!

• It is impossible to do this unless you have some idea as to how the elements of the infinite lists are generated – Jorge Fernández Hidalgo Mar 28 at 21:35

You can have two sets of numbers such that they never clash: For example, let both sets of numbers be $$N_1 = N_2 = \{1, 3, 5, \dots \}$$. Now, if you multiply $$N_2$$ by $$2$$, then the new set will be $$\{2, 6, 10, \dots \}$$ which cannot clash with the set $$N_1$$ (since $$N_1$$ only contains odd numbers, while $$N_2$$ contains even numbers.
Of course, this depends on the set of numbers $$N_1$$ and $$N_2$$ in question.