GWD-4-4滔滔

cournot

Q4:For a nonnegative integer n, if the remainder is 1 when 2ⁿ is divided by 3, then which of the following must be true?

I.                    n is greater than zero. n=0

II.                 3ⁿ = (-3)ⁿ

III.               √2ⁿ  is an integer.

 

A.         I only

B.          II only

C.         I and II

D.         I and III

E.          II and III

 这个题I 错，II and III怎么解？

lianxyz

If 2,n is an even;

if 3, n is an even ,too.

2^n= 3k +1 if n is an even.

Thus 2 and 3 are correct.

cournot

but how to get the answer that n is an even not an odd when 2^n=3k+1

kallyli

用数学归纳法给出证明，我怕英文表达不清，用中文啦

如果：2的2n次方除以3余1，此时2n为偶数， 则
2的2（n+1）次方除以3的余数等于2的2n次方的余数即1再乘以4，其除以3的余数为1，所以所有偶数幂时余数为1。
2的2n+1次方（此时幂为奇数），其余数等于2的2n次方的余数1再乘以2，余数为2，所以所有奇数幂时余数为2

用数归发证明很严谨，但是一我表达不清楚（不会表示次方），二没有必要，你带进去几个数字是一下就行了

[此贴子已经被作者于2006-7-12 12:22:06编辑过]

cournot

thanks a lot!

citicsbond

why is 1st choice not correct, as clearly mentioned "if the remainder is 1 when 2^n is divided by 3". assuming that n=0, then we could not get the remainder to be 1 !!

ppyy

? 2^0=1, 1/3不是余1么

citicsbond

ye...基础太差，惭愧！！

安静vs寂静

代入几个数字最快

确定一错了以后就没必要看2了  只看3就行了

christlulu

题做多了就想差了。。。晕，怎么老是这么粗心。。。