PREP 一道数学题求解

songxuebest

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

reddragon

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.

songxuebest

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

georgina1107

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

citidandan

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.