问题 解答题

试找出由0,1,2,3,4,5,6这7个数字组成的没有重复数字的七位数中,能被165整除的最大数和最小数(要求写出推理过程).

答案

∵165=3×5×11

∴此7位数必同时能被3,5,11整除,而0+1+2+3+4+5+6=21能被3整除,

∴排成的7位数只需能同时被11,5整除即可,

根据能被11整除的数的性质,设7位数奇位上的数字和为x,偶数位上的数字和为y,则

x-y是11的倍数而x-y与x+y的奇偶性相同,且x+y=21,

∴只有x-y=11或-11,即x=5,y=16或x=16,y=5

∵7位数能被5整除,

∴其末位数必为0或5,

当末位数必为0或5时,找不到4个数的和为5,

∴只有x=16,y=5,即该7位数的奇数位上的数字和为16,偶数位上的数字和为5,且其末位数必为5.

只有两组分法:

①、奇数位上的数字(1,4,6,5),偶数位上的数字(0,2,3)

②、奇数位上的数字(2,3,5,6)偶数位上的数字(0,1,4)

①中最大数和最小数分别为6342105和1042635;②中最大数和最小数分别为6431205和2031645

∴所求的最大数和最小数分别为6431205和1042635.

故答案为:6431205和1042635.

阅读理解

阅读短文,选择正确答案。

    An old farmer spent all his life on his farm in the country in the southeast of England. One day he

decided to visit the big city. Everything was new and strange to him, because he had never traveled to

the city before.

    The old farmer visited a lot of interesting places in the city.

    He rushed into a large tall building and saw a lift (电梯). When he watched, an old woman got into

the lift and closed the door. After a while, the door opened again and a very beautiful young girl came

out.

    The old farmer was very surprised, "What a funny little room!" He said to himself, "It's magic. It can

change an old woman into a young girl. The next time I come here, I'm going to bring my wife to take

the lift."

1. The old farmer lived             . [ ]

A. near the city

B. in the city

C. in the southeast of England

D. in a large tall building

2. The old farmer decided              one day. [ ]

A. to visit a building

B. to visit a city

C. to visit a factory

D. to visit a farm

3. The old farmer             . [ ]

A. came to the city for the first time

B. had traveled to the city several times

C. found nothing new and strange in the city

D. only visited the large tall building

4. When the old farmer was in a large tall building, he saw              get into the lift. [ ]

A. a young girl

B. an old man

C. his wife

D. an old woman

5. From the passage, we can guess the old farmer             . [ ]

A. knew the lift well

B. didn't know anything about the lift

C. wanted to buy a lift

D. didn't like the lift

报关编码