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While trying to understand what the competency floor will be for math teachers next year in middle school, I came across the question below on the skills test.

My question here is to clarify what is meant. Specifically, what is the difference between these two sets:

  • all positive integers
  • all positive integers writable in form $2^n * 3^m$

Original question: S is the set of all positive integers that can be written in the form $2^n * 3^n$, where n and m are positive integers. If a and b are two numbers in S, which of the following must also be in S ?

(assuming answers aren't relevant, question posted only for context)

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  • $\begingroup$ sorry last minute edit, second exponent changed to m $\endgroup$ – whitneyland Aug 24 '17 at 21:25
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    $\begingroup$ Is this a skills test for your daughter or for the teacher? As for your question: certainly the second set is a proper subset of the first. Not all integers can be expressed as the product of a power of 2 and a power of 3. $\endgroup$ – symplectomorphic Aug 24 '17 at 21:32
  • $\begingroup$ For the teacher. It turns out being a math teacher does not require a stem degree, let alone a math degree, rather just to pass a test on a few basic subjects. $\endgroup$ – whitneyland Aug 24 '17 at 21:35
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    $\begingroup$ The possible answer choices may in fact be *highly relevant". You should include them. $\endgroup$ – Bill Dubuque Aug 24 '17 at 23:54
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The number $5$ is in the first set, but not in the second one. Hence the two sets are not equal.

Edit : changed $4$ to $5$ after your edit

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  • $\begingroup$ wow, may have seemed trivial but thank you. apparently after a certain number of years without practice certain things become invisible. $\endgroup$ – whitneyland Aug 24 '17 at 21:33
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The second set is a proper subset of the first set. For example, $5$ is in the first set but not the second set.

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If $a \in S$ then $a = 2^n \times 3^m$ for positive $n$ and $m$. For example, $12=2^2 \times 3$. But $13 \notin S$ as $13 \neq 2^n \times 3^m$ for any positive integers $n$ and $m$.

Note that every element of $S$ is a positive integer. Therefore, if $a \in S$ then $a \in Z_+$, where $Z_+ = \{1,2,3,\cdots\}$ is the set of positive integers.

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you said $n$ and $m$ are positive integers, so everything that is not divisible by 6 is in $\mathbb{N}$ but not $S$.

This is not a full description of the complement of $S$, but should illuminate just how "small" $S$ is (even though it shares cardinality with $\mathbb{N}$)

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