请教3道prep1 的ps（7，19，27）

让我百思不得其解第七题

7.    905-!-item-!-187;#058&000575

For every positive even integer n, the function h(n) is defined to be the product of all the even integers from 2 to n, inclusive.  If p is the smallest prime factor of h(100) + 1, then p is

(A) between 2 and 10

(B) between 10 and 20

(C) between 20 and 30

(D) between 30 and 40

(E) greater than 40

19.   2463-!-item-!-187;#058&002761

For a finite有限的
                sequence
            顺序of nonzero numbers, the number of variations in sign is defined as the number of pairs of consecutive
            连续的terms of the sequence for which the product of the two consecutive terms is negative.  What is the number of variations in sign for the sequence 1, -3, 2, 5, -4, -6 ?

(A) One

(B) Two

(C) Three

(D) Four

(E) Five

所给的数如果考虑正负号的话，并没有连续的呀

27.   3330-!-item-!-187;#058&003295

38, 69, 22, 73, 31, 47, 13, 82

Which of the following numbers is greater than three-fourths of the numbers but less than one-fourth of the numbers in the list above?

(A) 56

(B) 68

(C) 69

(D) 71

(E) 73

题目就没咋看明白 什么叫“四分之三的以上数”，是以上数的什么？和？积？还是其他意思？

非常感谢nn解答

7.    905-!-item-!-187;#058&000575

For every positive even integer n对于每个正整数n, the function h(n) is defined to be the product of all the even integers from 2 to n函数h(n)定义为所有从2到n的偶数的积, inclusive包括n本身在内.  If p is the smallest prime factor of h(100) + 1如果p是h(100)+1的最小质因数, then p is求p

(A) between 2 and 10
(B) between 10 and 20
(C) between 20 and 30
(D) between 30 and 40
(E) greater than 40

解：h(100)+1=[(100!)/(1x3x5x...x99)]+1=[(1x2x3x4x5x...x100)+(1x3x5x...x99)]/(1x3x5x...x99)，可以观察出，这个除式里至少有质因数3,5...，因此p等于3，选A

19.   2463-!-item-!-187;#058&002761

For a finite sequence of nonzero numbers一个非零有限数列, the number of variations in sign is defined 其Variations of sign的数量定义为as the number of pairs of consecutive terms of the sequence这个数列中连续项的对数 for which the product of the two consecutive terms is negative要求这样的两个连续项的积为负.  What is the number of variations in sign for the sequence 1, -3, 2, 5, -4, -6 ?在1, -3, 2, 5, -4, -6这个数列中，Variation in sign的数量为几？

(A) One
(B) Two
(C) Three
(D) Four
(E) Five

所给的数如果考虑正负号的话，并没有连续的呀

解：在给出数列中，1x-3<0,-3x2<0,5x-4<0，满足条件的variations in sign为三对。C正确

注：不明白什么是variations in sign没关系，将它当作是一个未知数，题目中给出了这个未知数的定义。个人理解这个词是否可以翻译为“符号变量”？题中的consequtive是指数列中的连续项，而不是1,2,3...这个连续的意思。

27.   3330-!-item-!-187;#058&003295

38, 69, 22, 73, 31, 47, 13, 82
Which of the following numbers is greater than three-fourths of the numbers but less than one-fourth of the numbers in the list above? 如下哪个数比四分之三的上述数都大，而比四分之一的上述数要小？

(A) 56
(B) 68
(C) 69
(D) 71
(E) 73

题目就没咋看明白 什么叫“四分之三的以上数”，是以上数的什么？和？积？还是其他意思？

解：将上述8个数排序：13,22,31,38,47,69,"?",73,82。？位置的数即满足题意，因此正确答案为D。

**要注意读懂题的意思。如上个人观点，欢迎指正。

第一提我想了很久看到有人用!的方式解题点醒了我

2*4*6*8*....*100+1=2*(2*1)*(2*3)*(2*4)*...2*50+1

=(2^50)*50!+1

1到50所有的INTEGERS保证是2^50*50!的FACTOR

也就是说=2*某数=3*某数=4*某数.....

这时候再加个1上去自然无法被1到50任何数整除
            

所以FACTOR必定大于40

不知道这样对不对

我字认为是对了

谢谢nn们的解答，小女子豁然开朗

第一题应该是选E，我找到了往年的分析，小总结一下：

先将
            h(100) + 1= MingLiU","serif"; mso-fareast-font-family: simsun;">2*4*6*8*....*100+1=2*(2*1)*(2*3)*(2*4)*...2*50+1MingLiU","serif";">
        MingLiU","serif"; mso-fareast-font-family: simsun;">=(2^50)*50!+1MingLiU","serif";">
        MingLiU","serif"; mso-fareast-language: ZH-CN; mso-fareast-font-family: 宋体; mso-fareast-theme-font: minor-fareast;">

h(100) = MingLiU","serif"; mso-fareast-language: ZH-CN; mso-fareast-font-family: simsun;">(2^50)*50!MingLiU","serif"; mso-fareast-language: ZH-CN; mso-fareast-font-family: 宋体; mso-fareast-theme-font: minor-fareast;">

即MingLiU","serif";">1到MingLiU","serif";">50所有的整数都是h(100) 因素MingLiU","serif"; mso-fareast-font-family: simsun;">FACTOR

即h(100)是2，3，5，7，...43，47的倍数，其最大公因数为47

根据数理理论：如果一个整数m是整数n(非1)的倍数，那么m+1一定不是n的倍数

即h(n)的质因数都不可能是h(n)+1的质因数

例如：42=2*3*7，那么2，3，7都不可能是42+1的质因数

各位：

解：h(100)+1=[(100!)/(1x3x5x...x99)]+1=[(1x2x3x4x5x...x100)+(1x3x5x...x99)]/(1x3x5x...x99)，可以观察出，这个除式里至少有质因数3,5...，因此p等于3，选A

这个解法的错误在哪里呢，能否帮忙看一下。