smc

The number of positive integers under $1000$ that are divisible by $5$ is $\lfloor \frac{999}{5}\rfloor =199$. The number of positive integers under $1000$ that are divisible by $7$ is $\lfloor \frac{1000}{7}\rfloor =142$. Adding them together, we get $341$. However, we have over counted the numbers which are divisible by $35$. There are $\lfloor \frac{999}{35}\rfloor =28$ of these. So, the number of positive integers divisible be $7$ or $5$ under $1000$ is $341-28=313$. We can conclude that the number of positive integers divisible by neither $5$ or $7$ is $999-313=686$ or answer choice $\boxed{686}$.