Question

如果一个正整数的每一位数字都严格大于前一位数字，则称它为上坡整数。例如 $1357, 89$ 和 $5$ 都是上坡整数，但 $32, 1240$ 和 $466$ 不是。多少个上坡整数能被 $15$ 整除？

Call a positive integer an uphill integer if every digit is strictly greater than the previous digit. For example, $1357, 89,$ and $5$ are all uphill integers, but $32, 1240,$ and $466$ are not. How many uphill integers are divisible by $15?$

$4$

$5$

$6$

$7$

$8$

答案：C

知识点：子集整除性

难度评级：1480

视频讲解：

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文字解答：

能被 $15$ 整除的数，个位必须是 $0$ 或 $5$ 。若上坡整数的个位是 $0$ ，则这个数只能是 $0$ ，不是正整数。因此只需考虑个位为 $5$ 的数。

接着要求这个上坡整数是 $3$ 的倍数。其余数字只能从 $\{1,2,3,4\}$ 中选，所选集合的数字和除以 $3$ 必须余 $1$ 。数字 $3$ 取或不取不影响余数，所以先在不含 $3$ 的情况下计数再乘以 $2$ 。满足条件的子集只有 $3$ 个，即 $\{1\}, \{4\}$ 和 $\{1,2,4\}$ 。因此总共有 $6$ 个子集。

所以正确答案是 C。

If a number is divisible by $15,$ it has a units digit of $0$ or $5.$ If the units digit is $0$ and the digits are strictly increasing, then the number is $0,$ which isn't positive. Therefore, we can just look at numbers with a units digit of $5.$

Next, we need to find uphill integers that are a multiple of $3.$ This means the other digits are a subset of $\{1,2,3,4\}.$ Taking the sum of the set must have a remainder of $1$ when divided by $3.$ Also, having or taking out $3$ wouldn't affect the remainder, so we can take the number of subsets without a $3$ and multiply it by $2.$ There are only $3$ such subsets, namely $\{1\}, \{4\},$ and $\{1,2,4\}.$ Thus, there are $6$ total subsets.

Thus, the correct answer is C .

2021 AMC 10B Spring 第 16 题

答案：C

难度评级：1480

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