Solution:
Suppose $N$ is our integer. Then we have $N = \frac{6}{5} x = \frac{81}{100} y$ for some integers $x$ and $y$. In particular, $x$ is divisible by $5$ and $y$ is divisible by $100$. By multiplying by $100$, we have $120 x = 81 y$, and the smallest integers that satisfy this as well as the previous conditions are $x = 135$, $y = 200$. This yields $N = 162$.