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1994年全国高中联赛第二试第二题:将与105互素的所有正整数从小到大排成数列.试求出这个数列的第1000项.这道题目是要我们找出满足条件的第1000项是什么数?在现有的参考答案中都是构造数列来找的,其过程较复杂.本文给出一种“找”的新思路,即先将存在满足条件的1000个与105互素的数的范围“找到”,然后从大到小验证到与105互素时的数即为所求的数.具体解法如下:
In 1994, the second test of the National High School League Second Test: ranked all positive integers with 105 primes in small to large numbers. Try to find the 1000th item of this series. This topic is to find out the number of satisfying conditions. What is the number of 1000? In the existing reference answers are constructed to find the number of series, the process is more complicated. This paper gives a new way of “find”, that is, the first to meet the conditions of the 1000 and 105 each other The number of primes “find” and then verify from the largest to the smallest number when it is mutually prime with 105 is the number sought. The specific solution is as follows: