1 . 设
,其中,b为正奇数.定义数列
满足
,
.若正整数
,使得
为素数.证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a29fca134ddbac482c9821e78e6ab842.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a1f83d1a6f169d2e621b366431729888.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/91fc7b499561a3427ed1ef0354689f7e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/772e2d1d7398351c44c55366756f16ce.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0704f453b2de48d36911f7db496bbf82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/53500b98cf6e4620e77a8271c73549d3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76a3ac74cdb64183be40d576145a9cd5.png)
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2 . 数
被77除,所得的余数是.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b7d694b9d4b89a3588d789cbd27a2832.png)
A.45 | B.56 | C.67 | D.68 |
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3 . 设整数
,
,
,
满足
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e81e59019989b7dc2fb59b037ef6e010.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/18d463be2ce15e925c24b0929b7b0153.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96405d36641237835975484b8ed697e0.png)
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4 . 已知数列
满足
,
.
(1)证明:
为正整数数列;
(2)是否存在
,使得
?并说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b065334d8f60c49f4bd3d9f1373fe4cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac5f4a0a3f9437cd9ae4981f1bdf05e1.png)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(2)是否存在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f48c0161247f1a4624ebecdcf005bf05.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/29d98371cb419cce47ed6586c9bcb02e.png)
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5 . 设n为正整数.求满足以下条件的三元正整数组(a、b、c)的个数:
(ⅰ)ab=n;
(ⅱ)
;
(ⅲ)a、b、c的最大公约数为1.
(ⅰ)ab=n;
(ⅱ)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a60ae6d673752664497bf79647530e4f.png)
(ⅲ)a、b、c的最大公约数为1.
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2013高三·江苏·竞赛
6 . 若
除以3102的余数为1,则最小的正整数n为________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d73897c691c6329f9806397764c5b740.png)
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2013高三·江苏·竞赛
7 . 设p为奇素数,整数
均与p互素.若对
均有
,证明:
除以p的余数互不相同.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f01b3d5835c84b9ee45969f09de5b4cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/475860b9e1c51edaacd8b35f850a2939.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c37ee0117cd844570400494398a5a2aa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f01b3d5835c84b9ee45969f09de5b4cd.png)
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