GMAT prep上的一道数学题求质因数

RickySnow · 发表于 2013-2-17 16:45:54

For every positive 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 to 10
B.between 10 to 20
C.between 20 to 30
D.between 30 to 40
E.greater than 40
答案选E，对于这种找质因数的题目，总是感觉没有什么好的思路，东拼西凑的，请CD上的大大们求教一下~ 感激不尽~

RickySnow · 发表于 2013-2-17 17:04:36

自己顶下，求大神~

zyx20085 · 发表于 2013-2-17 17:19:44

As per definition, the value h(100) = 2 x 4 x 6 x 8 x ....... x 100

This can be simplified as:

h(100) = 2 x [1 x 2 x 3 x .......50] = 2 x 50!

If you add 1 to the above number we will get 2 x 50! + 1 which is an odd number. If you look at the expression 2 x (50!), the largest prime number factor of this number is 47 (because 47 is the largest prime factor that is less than 50). So if you add 1 to the above expression, then I think its prime factor has to be greater than 47. So the answer is (E).

RickySnow · 发表于 2013-2-17 17:42:23

As per definition, the value h(100) = 2 x 4 x 6 x 8 x ....... x 100

This can be simplified as:

h(100) = 2 x [1 x 2 x 3 x .......50] = 2 x 50!

If you add 1 to the above number we will get 2 x 50! + 1 which is an odd number. If you look at the expression 2 x (50!), the largest prime number factor of this number is 47 (because 47 is the largest prime factor that is less than 50). So if you add 1 to the above expression, then I think its prime factor has to be greater than 47. So the answer is (E).

-- by 会员 zyx20085 (2013/2/17 17:19:44)

首先谢谢解答，不过还有些疑问：首先题目要求的是求解最小质因数，你指出的是the largest prime number？；其次，因为A+1中的A里面包含有47就推断A+1的质因数大于40？个人觉得不太合理，比如说10*47+1=471，就有最小质因数3啊

cloudycici · 发表于 2013-2-17 18:00:06

lz你好像理解错了，得是2*（1*2*3.....*47*48*49*50)+1能被质数整除才可以，这个数是不可能被比47小（包含）47整除的；

RickySnow · 发表于 2013-2-17 18:29:48

好吧，还是没明白，姑且记住这个结论吧。。。
2*k!+1=pm, p大于比k小的最近一个质因数

谢谢各位~

carinabella · 发表于 2014-9-2 11:48:09

h(100)为什么会等于2*(50!)呢？
不应该等于(2^50)*(50!) 么？
嘛，不过这个细节不影响解题啦……

GMAT prep上的一道数学题求质因数

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