1 . 数列
满足:
是大于1的正整数,试证明:在数列
中存在相邻的两项,它们除以
余数相同.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57c4d9c843ed628701f262f3e80ccb62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/58e51235780886a13ff7ab8918e97d64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
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2023高三·全国·专题练习
2 . 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
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 . 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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5 . 已知素数
,
满足
.证明:存在正整数
使得
的十进制表示的各位数字之和是2或3.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1010846eeec6c9da29640f5aa3f8738.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9aa8a716a31b0f51b70fdf9bdb257909.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8870a6a0826d38a5bc06174e163bb392.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/971d064c23be0e9d1a29d99c2f9746aa.png)
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6 . 对每个正整数n,定义
为从1到n中所有与n不互质的正整数的和.求证:若
且
,则
是合数.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38fcec7af3520884b173b29bda6c657a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a9ba50ede8ef97b843accf839fef5c0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c7d9712c3b25f3030e166e136d3a4686.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ae9dd56551121a5b88be4dc6910c9ec0.png)
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7 . 正整数
,且
的素因子个数不超过2,对于任意整数
,若
,则有
成立,求证:
是质数.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0704f453b2de48d36911f7db496bbf82.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/2794218743c2cccfe26d67c803ca094b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d7c6588252b11da61d1bb7f9b7766a2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
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8 . 设
证明:对整数
,
,必有一个模4余1的素因子.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9beb3fc78a72d29aefd329ebd137cc80.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b5d0a73f50b3e4583f1c1b6d6bf0d18.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
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9 . 求所有的正整数n,使得方程
有正整数解.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c832ddf5af110f444d51b903ee2bf94.png)
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10 . 设
表示不超过n且与n互素的正整数的个数,g(n)满足对任意
,其中求和符号
表示d取遍n的所有正因子,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ea6957c794f2cbebe2380f26c3df877.png)
________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b3d40cb0f4dfbccdd4b6dadb06588fc8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4e53ba76575fa84cc608a76d7245ac2e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8c1a5bb06c1320240f5431b6b394caa4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ea6957c794f2cbebe2380f26c3df877.png)
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