求助两道官方prep题，刚模考完。。

PriSteve

Q1. P.S: For every positive even integer n, the fuction 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. 2<p<10
B. 10<p<20
c. 20<p<30
D. 30<p<40
E. p>40

Q2. D.S: Of the 60 animals on a certain farm, 2/3 are either pigs or cows. How many of the animals are cows?
(1). The farm has more than twise as many cows as it has pigs.
(2). The farm has more than 12 pigs.

Q1的答案是E，Q2的答案是C。 请大家教教我怎么做。。谢谢^^

eflet222

就是2,4,6,8每个都提一个2出来,就边成1到50的乘积再乘以50个2就得到了

PriSteve

嗯，第二题是我把more than twise 看掉了，我以为c=2p了。。谢谢你。
第一题的 h(100) can be rewritten as 2^50*(1*2*3*...*50) 是怎么得来的呢？ 我化简成了2^(1+2+...+50)

eflet222

第2题有人问过了,我简单说一下:
(1)+(2):
p>=13
c>2p

当p>=14时,c必然>28,c＋p一定大于４０
所以只有可能p＝１３,c＝２７这一种情况

PriSteve

谢谢

eflet222

Guest, this is definitely a difficult number properties question. Let's first consider the prime factors of h(100). According to the given function,
h(100) = 2*4*6*8*...*100

By factoring a 2 from each term of our function, h(100) can be rewritten as
2^50*(1*2*3*...*50).

Thus, all integers up to 50 - including all prime numbers up to 50 - are factors of h(100).

Therefore, h(100) + 1 cannot have any prime factors 50 or below, since dividing this value by any of these prime numbers will yield a remainder of 1.

Since the smallest prime number that can be a factor of h(100) + 1 has to be greater than 50, The correct answer is E.

Hope that helps
-Dan