Given a $101$-digit number $a$ and an arbitrary positive integer $b$. Prove that there is at most a $102$-digit positive integer $c$ such that any number of the form $\overline{caaa \dots ab}$ is composite.
Source: Saint Petersburg olympiad 2024, 11.4
Tags: number theory
Given a $101$-digit number $a$ and an arbitrary positive integer $b$. Prove that there is at most a $102$-digit positive integer $c$ such that any number of the form $\overline{caaa \dots ab}$ is composite.