2018 AMC 12A Problems/Problem 19

Problem

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

$\textbf{(A) } 16 \qquad \textbf{(B) } 17 \qquad \textbf{(C) } 19 \qquad \textbf{(D) } 23 \qquad \textbf{(E) } 36$

Solution 1

Note that the fractions of the form $\frac{1}{2^a3^b5^c},$ where $a,b,$ and $c$ are nonnegative integers, span all terms of the infinite sum.

Therefore, the infinite sum becomes $\begin{align*} \sum_{a=0}^{\infty}\sum_{b=0}^{\infty}\sum_{c=0}^{\infty}\frac{1}{2^a3^b5^c} &= \left(\sum_{a=0}^{\infty}\frac{1}{2^a}\right)\cdot\left(\sum_{b=0}^{\infty}\frac{1}{3^b}\right)\cdot\left(\sum_{c=0}^{\infty}\frac{1}{5^c}\right) \\ &= \frac{1}{1-\frac12}\cdot\frac{1}{1-\frac13}\cdot\frac{1}{1-\frac15} \\ &= 2\cdot\frac32\cdot\frac54 \\ &= \frac{15}{4} \end{align*}$ by a product of geometric series, from which the answer is $15+4=\boxed{\textbf{(C) } 19}.$

~athens2016 (Solution)

~MRENTHUSIASM (Revision)

Solution 2

Separate into $7$ separate infinite series's so we can calculate each and find the original sum:

The first infinite sequence shall be all the reciprocals of the powers of $2$ , the second shall be reciprocals of the powers of $3$ , and the third will consist of reciprocals of the powers of $5$ . We can easily calculate these to be $1, \frac{1}{2}, \frac{1}{4}$ respectively.

The fourth infinite series shall be all real numbers in the form $\frac{1}{2^a3^b}$ , where $a,b\geq1$ .

The fifth is all real numbers in the form $\frac{1}{2^a5^b}$ , where $a,b\geq1$ .

The sixth is all real numbers in the form $\frac{1}{3^a5^b}$ , where $a,b\geq1$ .

The seventh infinite series is all real numbers in the form $\frac{1}{2^a3^b5^c}$ , where $a,b,c\geq1$ .

Let us denote the first sequence as $a_{1}$ , the second as $a_{2}$ , etc. We know $a_{1}=1$ , $a_{2}=\frac{1}{2}$ , $a_{3}=\frac{1}{4}$ , let us find $a_{4}$ . factoring out $\frac{1}{6}$ from the terms in this subsequence, we would get $a_{4}=\frac{1}{6}(1+a_{1}+a_{2}+a_{4})$ .

Knowing $a_{1}$ and $a_{2}$ , we can substitute and solve for $a_{4}$ , and we get $\frac{1}{2}$ . If we do similar procedures for the fifth and sixth sequences, we can solve for them too, and we get after solving them $\frac{1}{4}$ and $\frac{1}{8}$ .

Finally, for the seventh sequence, we see $a_{7}=\frac{a_{8}}{30}$ , where $a_{8}$ is the infinite series the problem is asking us to solve for. The sum of all seven subsequences will equal the one we are looking for, so solving, we get $1+\frac{1}{2}+\frac{1}{4}+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{a_{8}}{30}=a_{8}$ , but when we separated the sequence into its parts, we ignored the $1/1$ , so adding in the $1$ , we get $1+1+\frac{1}{2}+\frac{1}{4}+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{a_{8}}{30}=a_{8}$ , which when we solve for, we get $\frac{29}{8}=\frac{29a_{8}}{30}$ , $\frac{1}{8}=\frac{a_{8}}{30}$ , $\frac{30}{8}=(a_{8})$ , $\frac{15}{4}=(a_{8})$ . So our answer is $\frac{15}{4}$ , but we are asked to add the numerator and denominator, which sums up to $\boxed{\textbf{(C) } 19}$ .

~~Edited by mprincess0229~~

Solution 3

This solution is Solution 1 but potentially clearer.

Clearly this is just summing over the reciprocals of the numbers of the form $2^i3^j5^k$ , where $i,j,k\in [0,\infty)$ . SO our desired sum is $\sum_{k=0}^{\infty}\sum_{j=0}^{\infty}\sum_{i=0}^{\infty}\frac{1}{2^i3^j5^k}$ . By the infinite geometric series formula, $\sum_{i=0}^{\infty}\frac{1}{2^i3^j5^k}$ is just $\frac{\frac{1}{3^j5^k}}{1-\frac{1}{2}}=\frac{2}{3^j5^k}$ . Applying the infinite geometric series formula again gives that $\sum_{j=0}^{\infty}\frac{2}{3^j5^k}=\frac{\frac{2}{5^k}}{1-\frac{1}{3}}=\frac{3}{5^k}$ . Applying the infinite geometric series formula again yields $\sum_{k=0}^{\infty}\frac{3}{5^k}=\frac{3}{1-\frac{1}{5}}=\frac{15}{4}$ . Hence our final answer is $15+4=\boxed{\textbf{(C) } 19}$ .

-vsamc

Video Solution by LetsSolveMathProblems

https://www.youtube.com/watch?v=woXlEargLpI&ab_channel=LetsSolveMathProblems

2018 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 18	Followed by Problem 20
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