Writing product and remainder Given some positive integers, an operation is to choose two integers $a\geq b$, delete them, and write $ab$ and $a\pmod b$ instead. Must the number $0$ eventually appear?
For example, when there are two numbers, the number $0$ must eventually appear. This is because one number keeps increasing and the other number decreasing. When there are more than two numbers, maybe a similar idea works (i.e. the numbers must keep getting "farther apart"), but I'm not sure how to formalize it.

If you have $(a_1,a_2,...,a_n)$ and you choose $a_i$ and $a_j$ with $i<j$, and replace $a_i$ with the modulo and $a_j$ with the product, you may notice that the modulo is less than the minimum of $a_i$ and $a_j$.
So the new tuple is smaller than the initial one via lexicographical order.
Now use that $\mathbb N^n$ is well ordered with lexicographical order. So the process cannot infinitely go on, and eventually a zero must appear.
