Question

Mary 想了一个正的两位数。她把它乘以 $3$ ，再加上 $11$ 。然后她交换所得结果的两个数字，得到一个介于 $71$ 和 $75$ 之间（含端点）的数。Mary 原来的数是多少？

Mary thought of a positive two-digit number. She multiplied it by $3$ and added $11.$ Then she switched the digits of the result, obtaining a number between $71$ and $75,$ inclusive. What was Mary's number?

$11$

$12$

$13$

$14$

$15$

答案：B

知识点：数字整除性逆推法

难度评级：960

解答：

交换数字后的结果在 $71,72,73,74,75$ 到 $17,27,37,47,57$ 之间，所以个位数字在 $6,16,26,36,46$ 到 $36\div3=12$ 之间，十位数字是。倒推交换数字这一步后，十位数字在到之间，个位数字是。再减去 $11$ ，十位和个位各减，得到十位在到、个位为 $6$ 的数。它必须是 $3$ 的倍数，所以只能是 $36$ 。再除以得。

所以正确答案是 B。

Work backward from the possible results. Reversing $71,72,73,74,75$ gives $17,27,37,47,57,$ respectively. Subtracting $11$ gives $6,16,26,36,46.$ Only $6$ and $36$ are divisible by $3,$ and only $36\div3=12$ is a two-digit number.

Thus, the correct answer is B .

2017 AMC 10B 第 1 题

答案：B

难度评级：960

解答：

其他年份的第 1 题