jmc

There are $20$ positive multiples of $5$ less than $101$. There are $14$ positive multiples of $7$ less than $101$. However, the least common multiple of $5$ and $7$ is $35$, and there are $2$ positive multiples of $35$ less than $101$. This means there are $20-2=18$ multiples of $5$ that aren't multiples of $7$, and $14-2=12$ multiples of 7 that aren't multiples of $5$, for a total of $18+12=\boxed{30}$.