Prep-T1-Q7 (PS)

[nomade]

7. 905-!-item-!-187;#058&000575
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

(A) between 2 and 10
(B) between 10 and 20
(C) between 20 and 30
(D) between 30 and 40
(E) greater than 40

答案E

請教一下這題的解題思路為何, 想了好久想不出個頭緒來
謝謝

[eced]

h(n)=2x4x6x8x....xn
h(100)+1=2x4x6x...x100+1=2^50*50! + 1

重點在那50!, 假如 小於50的質數是h(100)的factor,那就不會是h(100)+1的factor ,因為餘數會是1

[nomade]

謝謝eced的回答
式子是不是要寫成 h(100)+1 = 2(1*2*3....*50) = 2* 50!

還有是不是因為因為h(100)的因子裡面包含了1~50
所以不可能小於50啊?

[eced]

h(100)+1 = 2(1*2*3....*50) = 2* 50! +1
你上面式子寫錯了, 每個都能提出一個2 所以有50個2 =>2^50 , 2^50*50!才對
h(100)+1=2x4x6x...x100+1=2^50*50! + 1

假設h(100)+1最小因數是 n , n<50
例如47好了, 既然47是因數, 他就能整除h(100)+1
可是發現h(100)裡面有47的因數,可以整除 那h(100)+1被47除的餘數為1,表示47不是h(100)+1的因數
同理去想 小於50的其它因式 也是會一樣的結果
所以h(100)+1的因式是在 大於50外

[nomade]

Totally understand...

eced大大太感謝你了...
不厭其煩的回文章

[funbob]

好棒～感謝

[aqboy[origen]]

真強!!!