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Problem 1218

AMC 12 late, AIME early

Algebra Difficulty 5.0 Find the answer HMMT November

Let $S$ be the set of all positive integers whose prime factorizations only contain powers of the primes 2 and 2017 (1, powers of 2, and powers of 2017 are thus contained in $S$ ). Compute $\sum_{s \in S} \frac{1}{s}$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Official solution

Since every $s$ can be written as $2^{i} \cdot 2017^{j}$ for non-negative integers $i$ and $j$ , the given sum can be written as $\left(\sum_{i=0}^{\infty} \frac{1}{2^{i}}\right)\left(\sum_{j=0}^{\infty} \frac{1}{2017^{j}}\right)$ . We can easily find the sum of these geometric series since they both have common ratio of magnitude less than 1, giving us $\left.\left(\frac{1}{1-\frac{1}{2}}\right) \cdot \frac{1}{1-\frac{1}{2017}}\right)=\frac{2}{1} \cdot \frac{2017}{2016}=\frac{2017}{1008}$ .

Source: Omni-MATH, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.