1 . 取整函数被广泛应用于数论、函数绘图和计算机领域,其定义如下:设
,不超过x的最大整数称为x的整数部分,记作
,函数
称为取整函数.另外也称
是x的整数部分,称
为x的小数部分.
(1)直接写出
和
的值;
(2)设a,
,证明:
,且
,并求在b的倍数中不大于a的正整数的个数;
(3)对于任意一个大于1的整数a,a能唯一写为
,其中
为质数,
为整数,且对任意的
,
,i,
,称该式为a的标准分解式,例如100的标准分解式为
.证明:在
的标准分解式中,质因数
(
,
,
)的指数
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24a57996290794e082b21d8f1dfc322a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a2ab85825d4a002600ca41bd3cd2ee7d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b7e3204e4dc47a448860779349efcedf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a2ab85825d4a002600ca41bd3cd2ee7d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/575ee8b07f40772a9b87075cc609b116.png)
(1)直接写出
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0f49a0f96d2a4dcd830e826b86b00d3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/173c5f003f563fcbaadf6d39e4ddd8a6.png)
(2)设a,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08b5471b5cd78e6b6ad2bb4a29a5b23a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e820aee29ce5ad5b6fefc10d35278589.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d3051234cd772b983370427936e3a3f.png)
(3)对于任意一个大于1的整数a,a能唯一写为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/086b43dfd3e01b604f2dcded9e594eb6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fb5c607987b73502db63f77c9799f4bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/50a272adba0f1120109824440f0e252c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/52f2c7c9305b404f7363a376af101aa4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d5747eaf96a886203a5d58db42dfe30.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1d89c4ce427373fe50cf612d7e240422.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d72b2aa820c5ea9d988dd6d1cc44dae.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a8412f5256b2b370e421c07f18cc732.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fb5c607987b73502db63f77c9799f4bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83788fccd28d55365ec25a7170d69148.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10e468312d09c6563c9094b710a35a65.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48f093c61867ee4ce75f951d46b9b123.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42afc34ecc7d3dc70274fa77720d9b1a.png)
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