4. Let $r_{n}$ denote the remainder of the division of a natural number $n$ by 6. For which $n$ is the equality $r_{2 n+3}=r_{5 n+6}$ true?
4. Solution. The numbers $2 n+3$ and $5 n+6$ have the same remainder if their difference $5 n+6-(2 n+3)=3 n+3$ is divisible by 6, i.e., $3 n+3=6 k$. From this, we get $n=2 k-1, k \in \square$.