A positive integer is written on the board. Each second you write 3 or 9 on the right next to it. Is it true that you always get a composite number in finite time?
(One can easily check that the digits 0,1,2,4,5,6,7,8 easily give a composite number. That's why I am asking only for 3 and 9.)
If we only allowed 3, then given a prime $p\geq10$, the infinite sequence 3, 33, 333, ... will have two numbers with the same remainder mod $p$ (Pigeonhole principle) and by subtracting and removing 0s we get a number $A$ composed of 3s and divisible by $p$ - in which case $\overline{pA}$ shall be composite.
Any help appreciated!