2018 AMC 10A Problems/Problem 17

Problem

Let $S$ be a set of 6 integers taken from $\{1,2,\dots,12\}$ with the property that if $a$ and $b$ are elements of $S$ with $a<b$ , then $b$ is not a multiple of $a$ . What is the least possible values of an element in $S?$ $\textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 7$

Solution

Intuitively, one would see this list and start with prime numbers. However, there are only 5 prime numbers less than $12$ , making this impossible. It is also clear that another number can't be added in, so $2$ can't be the smallest. Next, we start the sequence with $3$ , and a bit of trial and error shows it's impossible. Lastly, starting with $4$ , we find that the sequence $4,5,6,7,9,11$ works, giving us $\boxed{\textbf{(C)} \text{ 4}}$ .

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2018 AMC 10A Problems/Problem 17

Problem

Solution