jmc

It's easy to count the number of three-digit numbers which are multiples of 5 or 7: the smallest multiple of 7 which is a three-digit number is $15\times 7=105$, and the largest multiple of 7 that is a three-digit number is $142\times 7=994$. Therefore, there are $142-15+1=128$ three-digit numbers that are multiples of 7. The smallest multiple of 5 that is a three-digit number is $20\times 5=100$, and the largest multiple of 5 that is a three digit number is $199\times 5=995$. So there are $199-20+1=180$ multiples of 5.

Now notice that we have counted some numbers twice: those multiples of $5\times 7=35$. The smallest multiple of 35 is $3\times 35=105$, the largest multiple of 35 is $28\times 35=980$. So there are $28-3+1=26$ multiples of 35.

We have 128 multiples of 7 and 180 multiples of 5, but we count 26 multiples twice. So, there are a total of $128+180-26=282$ distinct three-digit numbers that are multiples of 5 or 7 (or both). There are 900 three-digit numbers in total (from 100 to 999), so there are $900-282=\boxed{618}$ three-digit numbers that are not multiples of 7 nor 5.