名校
1 . 设a,b为非负整数,m为正整数,若a和b被m除得的余数相同,则称a和b对模m同余,记为
.
(1)求证:
;
(2)若p是素数,n为不能被p整除的正整数,则
,这个定理称之为费马小定理.应用费马小定理解决下列问题:
①证明:对于任意整数x都有
;
②求方程
的正整数解的个数.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c73aeb67aa5fa6797d0a68cfbf1a3d5.png)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bbfac455432b5ddc11bbbb62b165f1ef.png)
(2)若p是素数,n为不能被p整除的正整数,则
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/64b82d58ea4cb94ff8dc3aeb1c345a0e.png)
①证明:对于任意整数x都有
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/366bfef60e3b2c6fd95003cddbd66605.png)
②求方程
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8bc16a57919b711a9d34eed86b437f35.png)
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2024-02-27更新
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816次组卷
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5卷引用:河北省2024届高三下学期大数据应用调研联合测评(V)数学试题
河北省2024届高三下学期大数据应用调研联合测评(V)数学试题河北省沧州市泊头市大数据联考2024届高三下学期2月月考数学试题河北省秦皇岛市昌黎县开学联考2024届高三下学期开学考试数学试题(已下线)压轴题高等数学背景下新定义题(九省联考第19题模式)讲(已下线)新题型02 新高考新结构竞赛题型十五大考点汇总-2
解题方法
2 . 离散对数在密码学中有重要的应用.设
是素数,集合
,若
,记
为
除以
的余数,
为
除以
的余数;设
,
两两不同,若
,则称
是以
为底
的离散对数,记为
.
(1)若
,求
;
(2)对
,记
为
除以
的余数(当
能被
整除时,
).证明:
,其中
;
(3)已知
.对
,令
.证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1010846eeec6c9da29640f5aa3f8738.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f05bea470ae14b90937f6f71dc9a6242.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/99b2b0dcbc27df9950b26028e46f6c17.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e5865fd0fb7c35e8a4a1d311163290b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7cbe6ebc6c1d1a214f5ca478ae666cdb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1010846eeec6c9da29640f5aa3f8738.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b67a1f88ae28ecdb67c7f9c4ae61481b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dae890dd5b6300cf23b4905e86410317.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1010846eeec6c9da29640f5aa3f8738.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ff99d1615f90ff71b56ca1dfebd626d1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/420a12638f77a27c696f63ff946e8684.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2b0087ea124b6fd98fbbcb9bc4c2e09.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ce071bd0d6fa72ff4ba4e72d810d11f.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac54185ed8bb89c774ceb685408156c4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca7b54c31c5ab3831f260012758ffa12.png)
(2)对
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/099d389a1c0e5877350e62c52c4a724c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/28ab2ad5d8b72e3f26bef4be0697ec70.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3b6b09b60bf1bf8403c49bc17e365cd8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/77ebb2233e8492cf61fe9f9bc68af470.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3b6b09b60bf1bf8403c49bc17e365cd8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/77ebb2233e8492cf61fe9f9bc68af470.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/34fc26e532b65641a53eaa7e127aa683.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4d45dbe0a914249371aed3641515123.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/53ace23b21d7b119ad7ac5cf877c19f0.png)
(3)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ce071bd0d6fa72ff4ba4e72d810d11f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2793be26b839ae9f8f83cf2b5a597cd3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c1a6740a4f2378965bc019bc6aacd44a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f278b4fd6ed264265e3ccfac4ab7ef02.png)
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8卷引用:2024年1月普通高等学校招生全国统一考试适应性测试(九省联考)数学试题
2024年1月普通高等学校招生全国统一考试适应性测试(九省联考)数学试题(已下线)2024年1月普通高等学校招生全国统一考试适应性测试(九省联考)数学试题变式题16-192024年九省联考试卷分析及真题鉴赏(已下线)压轴题高等数学背景下新定义题(九省联考第19题模式)讲(已下线)微考点8-1 新高考新题型19题新定义题型精选(已下线)新题型02 新高考新结构竞赛题型十五大考点汇总-2(已下线)专题22 新高考新题型第19题新定义压轴解答题归纳(9大核心考点)(讲义)(已下线)专题8 考前押题大猜想36-40
2023高三·全国·专题练习
3 . 求证:
为任意整数
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2de16fdd14f9f111f1ec41514d234fa2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/04582116cd765fcc5a52f44279ad6c94.png)
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2023高三·全国·专题练习
4 . 已知素数
证明:
为整数,其中
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/78ede915be033958bbbc49cad146be35.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4af8969d81e3649f3c453758ff275340.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5e396e106da8f2ec43475cd0aa7257da.png)
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2023高三·全国·专题练习
5 . 若![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ea722d76386a47ab303d830d890ae6aa.png)
跑遍模
的简系,
跑遍模
的简系.证明:
跑遍模
的简系.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ea722d76386a47ab303d830d890ae6aa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c814128ea2139e33db94ea590e7c2223.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/77ab1256702aef4e9f1a5eb6c12ecc96.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aec19b68e3add9d5bfcc6269a1855b87.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1fbd67f60f04c278bdd867fdb3979dfb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5000a896e35177ba104a90a428061722.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/99e0a6a2bd38c553771889d0821fb868.png)
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2023高三·全国·专题练习
6 . 若
证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ea722d76386a47ab303d830d890ae6aa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5f1cecadde7302929cb3e1117f0625de.png)
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2023高三·全国·专题练习
7 . 设
是正整数,
(1)证明:![](https://staticzujuan.xkw.com/quesimg/Upload/formula/182c38174d9345986d8cb38cdf8217a4.png)
(2)证明:![](https://staticzujuan.xkw.com/quesimg/Upload/formula/97f14e67260b7ce9fb9a1fba4d23e7b3.png)
(3)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/182c38174d9345986d8cb38cdf8217a4.png)
(2)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/97f14e67260b7ce9fb9a1fba4d23e7b3.png)
(3)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/847e7fba99adadf43d8429c2598823be.png)
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2023高三·全国·专题练习
解题方法
8 . 已知数列
满足
.
(1)证明:
是正整数数列;
(2)是否存在
,使得
?并说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63d471926f7b27322d90c82b9ce21d3d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aaa39c95c48104697357b8f0e98c9840.png)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63d471926f7b27322d90c82b9ce21d3d.png)
(2)是否存在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24f8494594299d0ecce6e1e52151f402.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5d3ca83b04f476083c4044c33347a038.png)
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2023高三·全国·专题练习
9 . 已知数列
满足
.
(1)证明:
是正整数数列;
(2)是否存在
,使得
?并说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63d471926f7b27322d90c82b9ce21d3d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df3f0a8c6cf53d0b0b71ccf8e6f5226a.png)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63d471926f7b27322d90c82b9ce21d3d.png)
(2)是否存在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bc466f4022d38bf434544641fae2b51c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c4122bdc1b2a61097735c85b2dd655c.png)
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2023高三·全国·专题练习
10 . 意大利数学家斐波那契的《算经》中记载了一个有趣的数列:
,
,
,
,
,
,
,
,
,
,
,
,
,这就是著名的斐波那契数列,该数列的前
项中奇数的个数为_______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdaa19de263700a15fcf213d64a8cd57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdaa19de263700a15fcf213d64a8cd57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61128ab996360a038e6e64d82fcba004.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ca7d1107389675d32b56ec097464c14.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d91e07104b699c4012be2d26160976a2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd3304e23f3b0f9569c4140ca89b6498.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5453184e251cfe787b5965cd38426962.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ea1834490aacbee800ed5721312f4be1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3fd1a0475c383701348a36c35aea32f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/92128c6c226ce688bc160fb86854f2fc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/796e3e32d29652fb3ed9984c414aec35.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/459cc36d02d31815ba9f2d05e130e0cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/07096af3b99fd1cb11c31f19a2c6408e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d01dd350dc95f42f1883e0cc7aae084.png)
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