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标题: (2001 年，北京初中数学竞赛题) 1和0交替排列，组成下面形式的一串数 [打印本页]

作者: 野菜花 时间: 2005-2-6 23:48
标题: (2001 年，北京初中数学竞赛题) 1和0交替排列，组成下面形式的一串数

101, 10101, 1010101, 101010101, ...

问在这串数中有多少个质数？并请证明你的结论

作者: fzy 时间: 2005-2-7 03:44
标题: 回复：(2001 年，北京初中数学竞赛题) 1和0交替排列，组成下面形式的一串数

Only 1. We know 101 is prime. For any other (call it n), n * 11 = 11....11 (2k 1's for some k > 2) = 11...11 * 100...001.(First factor contains k 1's, second k-1 0's.) Now one of them is divisible by 11, and the other devides n.

作者: 野菜花 时间: 2005-2-7 03:56
标题: Great! [:B]

Great!

作者: fzy 时间: 2005-2-8 22:40
标题: Thanks for the question

Thanks for the question. It gave me the idea to solve this problem in WXC:

The repeat of a positive integer is obtained by writing it twice in a row (so, for example, the repeat of 254 is 254254). Is there a positive integer whose repeat is a perfect square?

作者: y 时间: 2005-2-9 02:15
标题: Why?

Could you explain more? We need to show a factor in n. I do not see your reason for it.

作者: y 时间: 2005-2-9 08:10
标题: I got it

I got it

作者: 野菜花 时间: 2005-2-11 01:59
标题: Your question is interesting, why don't you post

it here as 加新贴？

作者: fzy 时间: 2005-2-11 17:47
标题: I will

All right. I did not because I thought people here like only the quick-wit type of problems.

作者: 小牛牛 时间: 2009-6-29 17:02
标题: 回复：(2001 年，北京初中数学竞赛题) 1和0交替排列，组成下面形式的一串数

101一个

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