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 $3$ 。求的值。 $\boxed{\boxed{11}\times\boxed{20}}.$

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

解答：

$11$ 是质数，所以它只有 $2$ 个正因数。

$20$ 的质因数分解是因数个数等于各指数加一后的乘积。 $2^2 \cdot 5.$

这里得到 $(2 + 1) (1 + 1) = 3 \cdot 2 = 6.$

因此里面的乘积是 $2 \cdot 6 = 12$ 。而 $12$ 的质因数分解是它也有个因数。 $2^2 \cdot 3.$ $(2 + 1) (1 + 1) = 3 \cdot 2 = 6$

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

Thus, A is the correct answer.

1997 AMC 8 第 11 题

答案：A

难度评级：1140

解答：

其他年份的第 11 题