15) How many positive integers are multiples of at least one of 5 and 7 and less than or equal to 1000?
(A) 288,
(B) 302,
(C) 314
(D) 342,
(E) 382.
15. The answer is $(\mathbf{C})$.

We need to consider the multiples of 5 less than or equal to 1000, which are 200; to these we must add the multiples of 7 less than or equal to 1000, which are 142. We get $200+142=342$. From these, we must subtract the numbers that are both multiples of 5 and 7 (because these have been counted twice): these are all and only the multiples of 35 less than or equal to 1000, which are 28. Overall, the numbers we are looking for are $342-28=314$.