Question

小于 $2020$ 的正偶数且为 $3$ 的倍数的数中，有多少个是完全平方数？

How many positive even multiples of $3$ less than $2020$ are perfect squares?

$7$

$8$

$9$

$10$

$12$

答案：A

知识点：完全平方数整除性

难度评级：960

解答：

一个数既是偶数又是 $3$ 的倍数时，它是 $6$ 的倍数。若它还是完全平方数，则其平方根必须同时被 $2$ 和 $3$ 整除，因而被 $6$ 整除。因此所求平方数恰好为，其中 $k$ 为正整数。 $(6k)^2<2020$

因为 $42^2=1764<2020$ 且 $48^2=2304>2020$ ，所以 $k=1,2,\ldots,7$ ，共有 $7$ 个数。

所以正确答案是 A。

A number that is both even and a multiple of $3$ is a multiple of $6.$ If such a number is also a perfect square, its square root must be divisible by both $2$ and $3,$ hence by $6.$ Therefore the numbers counted are exactly $(6k)^2<2020$ for positive integers $k.$

Since $42^2=1764<2020$ and $48^2=2304>2020,$ we have $k=1,2,\ldots,7,$ for a total of $7$ numbers.

Thus, A is the correct answer.

2020 AMC 10B 第 7 题

答案：A

难度评级：960

解答：

其他年份的第 7 题