2000 AMC 10 第 11 题

Question

从

4

和

18

之间选出两个不同的质数。用它们的乘积减去它们的和，可能得到下列哪个数？

Two different prime numbers between

4

and

18

are chosen. When their sum is subtracted from their product, which of the following numbers could be obtained?

21

60

119

180

231

答案：C

知识点：质数奇偶性极限情形界定

难度评级：1370

小提示：

4

和

18

之间的质数是

5

、

7

、

11

、

13

、

17

，它们都是奇数。

The primes between

4

and

18

are

5,

7,

11,

13,

17,

all odd

大提示：

注意

xy - (x + y)

= (x - 1)(y - 1) - 1

，它是奇数，并且随着

x

、

y

变大而增大。

Note that

xy - (x + y)

= (x - 1)(y - 1) - 1,

which is odd and increases with

x

and

y

解答：

在

4

和

18

之间的质数是

5

、

7

、

11

、

13

和

17

。两个奇数的乘积是奇数、和是偶数，所以

xy - (x + y)

是奇数。

又因为

xy - (x + y)

= (x - 1)(y - 1) - 1

会随着任一质数增大而增大，结果从

5 \cdot 7 - 12 = 23

到

13 \cdot 17 - 30 = 191

。

选项中唯一在

[23, 191]

内的奇数是

119 = 11 \cdot 13 - (11 + 13)

。

所以正确答案是 C。

The primes between

4

and

18

are

5,

7,

11,

13,

and

17.

The product of two of them is odd and the sum is even, so

xy - (x + y)

is odd.

Since

xy - (x + y)

= (x - 1)(y - 1) - 1

increases as either prime increases, the result ranges from

5 \cdot 7 - 12 = 23

up to

13 \cdot 17 - 30 = 191.

The only odd option in

[23, 191]

is

119 = 11 \cdot 13 - (11 + 13).

Thus, the correct answer is C.

2000 AMC 10 第 11 题

答案：C

难度评级：1370

小提示：

大提示：

解答：

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