Consider all $100$-digit numbers divisible by $19$. Prove that the number of such numbers not containing the digits $4, 5$, and $6$ is the number of such numbers that do not contain the digits $1, 4$ and $7$.
Source: All-Russian MO 2023 Final stage 9.6
Tags: number theory
Consider all $100$-digit numbers divisible by $19$. Prove that the number of such numbers not containing the digits $4, 5$, and $6$ is the number of such numbers that do not contain the digits $1, 4$ and $7$.