PREP 一道数学题求解

shiropetto729 · 发表于 2010-12-3 02:54:21

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

shiropetto729 · 发表于 2010-12-3 02:55:24

只顾着打题，忘了说谢谢各位帮忙啦！！！

sdcar2010 · 发表于 2010-12-3 04:47:49

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.

shiropetto729 · 发表于 2010-12-3 05:59:10

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

amcv · 发表于 2010-12-5 17:34:15

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

sdcar2010 · 发表于 2010-12-5 20:42:56

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 (all the even number from) 2 to n 吗？

谢谢

-- by 会员 amcv (2010/12/5 17:34:15)

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.

gesteven · 发表于 2011-1-10 07:30:18

Thanks for the answer~~

PREP 一道数学题求解

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