Question

设 $A$ 为质因数只有 $2$ 、 $3$ 或 $5$ 的正整数集合。所有 $A$ 中元素倒数的无穷和可表示为 $\tfrac{m}{n}$ ，其中 $m$ 和 $n$ 是互质的正整数。 $m + n$ 是多少？ $\begin{aligned} &\frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} \\ &\quad {}+ \frac{1}{5} + \frac{1}{6} + \frac{1}{8} + \frac{1}{9} \\ &\quad {}+ \frac{1}{10} + \frac{1}{12} + \frac{1}{15} + \frac{1}{16} \\ &\quad {}+ \frac{1}{18} + \frac{1}{20} + \cdots \end{aligned}$

Let $A$ be the set of positive integers that have no prime factors other than $2,$ $3,$ or $5.$ The infinite sum $\begin{aligned} &\frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} \\ &\quad {}+ \frac{1}{5} + \frac{1}{6} + \frac{1}{8} + \frac{1}{9} \\ &\quad {}+ \frac{1}{10} + \frac{1}{12} + \frac{1}{15} + \frac{1}{16} \\ &\quad {}+ \frac{1}{18} + \frac{1}{20} + \cdots \end{aligned}$ of the reciprocals of all the elements of $A$ can be expressed as $\tfrac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. What is $m + n?$

$16$

$17$

$19$

$23$

$36$

答案：C

知识点：等比数列求和质因数分解

难度评级：1930

解答：

$A$ 中每个元素都可唯一写成 $2^i 3^j 5^k$ ，其中 $i, j, k \ge 0$ ，所以所有倒数的和可分解为这等于 $2 \cdot \tfrac32 \cdot \tfrac54 = \tfrac{15}{4}$ 。因为 $\gcd(15, 4) = 1$ ，所以 $m + n = 15 + 4 = 19$ 。 $\begin{aligned} &\left(\sum_{i \ge 0} \tfrac{1}{2^i}\right) \\ &\quad {}\cdot \left(\sum_{j \ge 0} \tfrac{1}{3^j}\right) \\ &\quad {}\cdot \left(\sum_{k \ge 0} \tfrac{1}{5^k}\right) \\ &= \frac{1}{1 - \frac12} \cdot \frac{1}{1 - \frac13} \\ &\quad {}\cdot \frac{1}{1 - \frac15}. \end{aligned}$

所以正确答案是 C。

Each element of $A$ is uniquely $2^i 3^j 5^k$ with $i, j, k \ge 0,$ so summing all reciprocals factors as $\begin{aligned} &\left(\sum_{i \ge 0} \tfrac{1}{2^i}\right) \\ &\quad {}\cdot \left(\sum_{j \ge 0} \tfrac{1}{3^j}\right) \\ &\quad {}\cdot \left(\sum_{k \ge 0} \tfrac{1}{5^k}\right) \\ &= \frac{1}{1 - \frac12} \cdot \frac{1}{1 - \frac13} \\ &\quad {}\cdot \frac{1}{1 - \frac15}. \end{aligned}$ This equals $2 \cdot \tfrac32 \cdot \tfrac54 = \tfrac{15}{4}.$ With $\gcd(15, 4) = 1,$ $m + n = 15 + 4 = 19.$

Thus, the correct answer is C.

2018 AMC 12A 第 19 题

答案：C

难度评级：1930

解答：

其他年份的第 19 题