1997 AMC 8 第 11 题

Question

令

\boxed{N}

表示

N

的正整数因数个数。例如，

\boxed{3}=2

，因为

3

有两个因数

1

和

3

。求下式的值：

\boxed{\boxed{11}\times\boxed{20}}\text{。}

Let

\boxed{N}

mean the number of whole number divisors of

N.

For example,

\boxed{3}=2

because

3

has two divisors,

1

and

3.

Find the value of

\boxed{\boxed{11}\times\boxed{20}}.

6

8

12

16

24

答案：A

知识点：因数个数质因数分解

难度评级：1140

小提示：

用质因数分解来数因数个数

Use prime factorization to count divisors

大提示：

先计算里面的方框，再计算外面的方框

Evaluate the inside boxes before the outside box

解答：

11

是质数，所以它只有

2

个正因数。

20

的质因数分解是

2^2 \cdot 5\text{。}

一个数的因数个数等于其质因数分解中各指数加一后的乘积。

这里得到

(2 + 1) (1 + 1) = 3 \cdot 2 = 6\text{。}

因此里面的乘积是

2 \cdot 6 = 12

。而

12

的质因数分解是

2^2 \cdot 3\text{，}

所以它也有

(2 + 1) (1 + 1) = 3 \cdot 2 = 6

个因数。

所以正确答案是 A。

We know that

11

is prime, which means that it only has

2

divisors.

The prime factorization of

20

is

2^2 \cdot 5.

Recall that the number of divisors a number has is the product of all the exponents plus one in the prime factorization.

Here, that product would be

(2 + 1) (1 + 1) = 3 \cdot 2 = 6.

Then

2 \cdot 6 = 12.

We have the prime factorization of

12

is

2^2 \cdot 3.

This also has

(2 + 1) (1 + 1) = 3 \cdot 2 = 6