Competition problem — Number theory

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Number theory Difficulty 5.9 AIME, harder Find the answer

1. (i) Find the number of integers from 1 to 2000 that are not divisible by 10, 14, or 21;
(ii) Find the number of integers from 1 to 1000 that are divisible by 3 and 7, but not by 5;
(iii) Find the number of integers from 1 to 1000 that are divisible by 3 or 7, but not by 5;
(iv) Find $\pi(N)$ , for $N=200,300,400,500,600,700,800,900,1000$ ;
(v) Find the number of integers from 2 to 1000 whose prime factors are all greater than 17;
(vi) Find the number of integers from 2 to 200 whose prime factors are all greater than 5 but not greater than 17.

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

$\begin{array}{l}\text { 1. (i) } 1638 \text {; (ii) } 47-9=38 \text {; (iii) } 333+142-66-28+38=419 \text {; (iv) } 46,62 \text {, } \\ 78,95,109,125,139,154,168 \text {; (v) } 168-7+7+5+1=174 \text {; (vi) } 11 \text {. }\end{array}$

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.