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Problem 783

AMC 12 late, AIME early

Number theory Difficulty 4.3 Find the answer HMMT November · United States

Let $S$ be the set of all positive integers whose prime factorizations only contain powers of the primes $2$ and $2017$ (that is, $1$ , 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

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
$\left(\frac{1}{1-\frac{1}{2}}\right) \cdot \frac{1}{1-\frac{1}{2017}} = 2 \cdot \frac{2017}{2016} = \frac{2017}{1008}.$

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