Question

Josh 写下数字 $1,2,3,\ldots,99,100$ 。他划掉 $1$ ，跳过下一个数 $(2)$ ，划掉 $3$ ，并继续交替跳过和划掉直到列表末尾。然后他回到列表开头，划掉第一个剩下的数 $(2)$ ，跳过下一个数 $(4)$ ，划掉 $6$ ，跳过 $8$ ，划掉 $10$ ，如此继续到末尾。Josh 以这种方式继续，直到只剩一个数。这个数是多少？

Josh writes the numbers $1,2,3,\ldots,99,100.$ He marks out $1,$ skips the next number $(2),$ marks out $3,$ and continues skipping and marking out the next number to the end of his list. Then he goes back to the start of his list, marks out the first remaining number $(2),$ skips the next number $(4),$ marks out $6,$ skips $8,$ marks out $10,$ and so on to the end. Josh continues in this manner until only one number remains. What is that number?

$13$

$32$

$56$

$64$

$96$

答案：D

知识点：2的幂找规律

难度评级：1440

解答：

第一轮划掉奇数，留下 $2$ 的倍数。第二轮划掉 $2,6,10,\ldots$ 留下 $4$ 的倍数。一般地，第 $n$ 轮后只剩 $2^n$ 的倍数。最后幸存的是不超过 $100$ 的最大 $2$ 的幂，即 $2^6=64$ 。

所以正确答案是 D。

The first pass removes the odd numbers, leaving the multiples of $2.$ The second pass removes $2,6,10,\ldots,$ leaving the multiples of $4.$ In general, after the $n$ th pass only the multiples of $2^n$ remain. The surviving number is the highest power of $2$ not exceeding $100,$ which is $2^6=64.$

Thus, the correct answer is D.

2016 AMC 12B 第 7 题

答案：D

难度评级：1440

解答：

其他年份的第 7 题