How many three-digit numbers are multiples of neither 5 nor 7?
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 $5\times7=35$. The smallest multiple of 35 is $3\times 35 = 105$, the largest multiple of 35 is $28\times35 =980$. 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.