7. Find the smallest positive integer that is $20 \%$ larger than one integer and $19 \%$ smaller than another.
Answer: 162
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 $N=162$.