8. A positive integer $T$ is said to be triangular if $T=1+2+\cdots+n$ for some positive integer $n$. Find the smallest positive integer $k$ such that whenever $T$ is triangular, $81 T+k$ is also triangular.
Answer: 10
Solution: Clearly, taking $T=1$, we must have $k \geq 10$. We show that $k=10$ indeed works. If $T$ is triangular, then $T=\frac{n(n+1)}{2}$ for some positive integer $n$. Then, we have
$$
\begin{aligned}
81 T+10 & =\frac{81 n(n+1)}{2}+10 \\
& =\frac{81 n^{2}+81 n+20}{2} \\
& =\frac{(9 n+4)(9 n+5)}{2}
\end{aligned}
$$
which indeed is triangular as well.