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标题: PREP 一道数学题求解 [打印本页]

作者: pangpangshin 时间: 2011-2-11 21:29 标题: PREP 一道数学题求解

1，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
1) between 2 and 10
2)between 10 and 20
3)between 20 and 30
4)between 30 and 40
5)greater than 40

Answer is 5)greater than 40

作者: abixiaoxinyue 时间: 2011-2-12 07:34

This one has been explained before.

Basically,
1) These consecutive natural numbers h(100) and [h(100) + 1] are co-prime, meaning that they do not share any prime numbers as their cofactors.
2) h(100) contains all the prime numbers between 2 (from 2) and 47 (from 94).
3) Then prime numbers among factors of [h(100) + 1] would not include any prime numbers between 2 and 47.
4) Then the smallest prime facotr of [h(100) + 1] is bigger than 47.

作者: avalon9527 时间: 2011-2-12 07:48

h(n)=2*4*6*...*100=2的50次方*(1*2*3*4*5*....*50)
由此可以看出，从2到50均能被h(n)整除，而
h(n)+1=2的50次方*(1*2*3*...*50)+1
也就是说从2到50均被h(n)+1除余数为1
所以其质因子不可能是2到50之内的。

作者: pangpangshin 时间: 2011-2-12 18:42

Hey thank you so much! I got it right now. Really appreciate your answer.

作者: xiexiang 时间: 2011-2-13 06:54

2) h(100) contains all the prime numbers between 2 (from 2) and 47 (from 94).

请问，这一步有些不理解，原文不是说 h（100）是 product of 2 to n 吗？

谢谢

作者: abixiaoxinyue 时间: 2011-2-14 20:45

2, 4, 6, 8, 10, 12, 14, 16, 18, . . . 92, 94, 96, 98, 100
is the same as
2, 2*2, 2*3, 2*4, 2*5, 2*6, 2*7, 2*8, 2*9, . . . 2*46, 2*47, 2*48, 2*49, 2*50
So the product of the second series contains factors of 2 and 2, 3, 4 ,5, 6, 7, 8, 9, . . . 46, 47, 48, 49, 50.
47 is the biggest factor of h(100), which is a prime number.

作者: pangpangshin 时间: 2011-2-15 06:53

Thanks for the answer~~

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