AMC10 2021 B

AMC10 2021 B · Q16

AMC10 2021 B · Q16. It mainly tests Basic counting (rules of product/sum), Divisibility & factors.

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 $15$?

称一个正整数为上坡整数，如果每个数字严格大于前一个数字。例如，$1357, 89,$ 和 $5$ 都是上坡整数，但 $32, 1240,$ 和 $466$ 不是。有多少个上坡整数能被 $15$ 整除？

(A) 4 4

(B) 5 5

(D) 7 7

(E) 8 8

Answer

Correct choice: (C)

正确答案：（C）

Solution

The divisibility rule of $15$ is that the number must be congruent to $0$ mod $3$ and congruent to $0$ mod $5$. Being divisible by $5$ means that it must end with a $5$ or a $0$. We can rule out the case when the number ends with a $0$ immediately because the only integer that is uphill and ends with a $0$ is $0$ which is not positive. So now we know that the number ends with a $5$. Looking at the answer choices, the answer choices are all pretty small, so we can generate all of the numbers that are uphill and are divisible by $3$. These numbers are $15, 45, 135, 345, 1245, 12345$, or $\boxed{\textbf{(C)} ~6}$ numbers.

$15$ 的整除规则是该数必须模 $3$ 同余 $0$ 且模 $5$ 同余 $0$。能被 $5$ 整除意味着它必须以 $5$ 或 $0$ 结尾。我们可以立即排除以 $0$ 结尾的情况，因为唯一以 $0$ 结尾且为上坡的整数是 $0$，它不是正整数。所以现在我们知道该数以 $5$ 结尾。查看答案选项，答案选项都很小，所以我们可以生成所有上坡且能被 $3$ 整除的数。这些数是 $15, 45, 135, 345, 1245, 12345$，共 $\boxed{\textbf{(C)} ~6}$ 个数。

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