问题 解答题

能够找到这样的四个正整数,使得它们中任两个数的积与2002的和都是完全平方数吗?若能够,请举出一例;若不能够;请说明理由.

答案

偶数的平方能被4整除,奇数的平方被4除余1,即正整数的平方被4除余0或1.

若存在正整数满足ninj+2002=m2;i,j=1,2,3,4,n是正整数;

∵2002被4除余2,

∴ninj被4除应余2或3.

(1)若正整数n1,n2,n3,n4中有两个是偶数,

设n1,n2是偶数,则n1n2+2002被4除余2,与正整数的平方被4除余0或1不符,

故正整数n1,n2,n3,n4中至多有一个是偶数,至少有三个是奇数.

(2)在这三个奇数中,被4除的余数可分为余1或3两类,

根据抽屉原则,必有两个奇数属于同一类,

则它们的乘积被4除余1,与ninj被4除余2或3的结论矛盾.

综上所述,不能找到这样的四个正整数,使得它们中任两个数的积与2002的和都是完全平方数.

阅读理解

I was in my third year of teaching creative writing at a high school in New York, when one of my students, 15-year-old Mikey, gave me a note from his mother. It explained his absence from class the day before.

I had seen Mikey himself writing the note at his desk. Most parental-excuse notes I received were penned by my students. If I were to deal with them, I’d be busy 24 hours a day.

The forged excuse notes made a large pile, with writing that ranged from imaginative to crazy. The writers of those notes didn’t realize that honest excuse notes were usually dull: “Peter was late because the alarm clock didn’t go off.”

Isn’t it remarkable, I thought, how the students complained and said it was hard putting 200 words together on any subject? But when they produced excuse notes, they were brilliant.

So one day I typed out a dozen excuse notes and gave them to my classes. I said, “They’re supposed to be written by parents, but actually they are not. True, Mikey?” The students looked at me nervously.

“Now, this will be the first class to study the art of the excuse note --- the first class, ever, to practice writing them. You’re so lucky to have a teacher like me who has taken your best writing and turned it into a subject worthy of study. ”

Everyone smiled as I went on, “You used your imaginations. So try more now. Today I’d like you to write ‘An Excuse Note from Adam to God’ or ‘An Excuse Note from Eve to God’.” Heads went down. Pens raced across paper. For the first time ever I saw students so careful in their writing that they had to be asked to go to lunch by their friends.

The next day everyone had excuse notes. Heated discussions followed. The headmaster entered the classroom and walked up and down, looking at papers, and then said, “I’d like you to see me in my office.”

When I stepped into his office, he came to shake my hand and said, “I just want to tell you that that lesson, that task, whatever the hell you were doing, was great. Those kids were writing on the college level. Thank you. ”

小题1:What did the author do with the students found dishonest?

A.He reported them to the headmaster.

B.He lectured them hard on honesty.

C.He had them take notes before lunch.

D.He helped improve their writing skills.小题2:The author found that compared with the true excuse notes, the produced ones by the students were usually__________.

A.less impressive

B.more imaginative

C.worse written

D.less convincing小题3:The author had the students practice writing excuse notes so that the students could learn_________.

A.the importance of being honest

B.how to write excuse notes skillfully

C.the pleasure of creative writing

D.how to be creative in writing小题4:The underlined word “forged” in the second paragraph means “______”.

A.former

B.copied

C.false

D.honest小题5:What did the headmaster think of the author’s way of teaching?

A.Effective.

B.Difficult

C.Misleading.

D.Reasonable

单项选择题