$32 \cdot 28$ is less than 1000 and the number of positive integers that cannot be divided by 5 and 7 is
(A) 688.
(B) 686.
(C) 684.
(D) 658.
(E) 630.
(17th American High School Mathematics Examination, 1966)
[Solution] If $[x]$ represents the greatest integer not exceeding $x$, note that:
the number of positive integers less than $n$ (a natural number) is $n-1$;
the number of positive integers less than $n$ and divisible by $d$ does not exceed $\left[\frac{n-1}{d}\right]$.
Thus, among the 999 positive integers less than 1000, the number of positive integers that are not divisible by 5 and 7 is
$$
999-\left[\frac{999}{5}\right]-\left[\frac{999}{7}\right]+\left[\frac{999}{5 \cdot 7}\right]=999-199-142+28=686 \text {. }
$$

Note that the positive integers divisible by $5 \cdot 7$ are subtracted twice. Therefore, the answer is $(B)$.