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Bob: I hear you won the lottery.

Alice: So I did!

Bob: What six numbers did you win it with?

Alice: Can't remember. All I recall is that they were all different, and none greater than 28.

Bob: Anything else?

Alice: Yes. I checked that no other six different numbers, none greater than 28, have the same common factor graph as mine.

(The common factor graph of a set of integers is the graph whose vertices are the integers, two of which are joined by an edge if they have a common factor greater than 1.)

What six numbers did Alice win the lottery with?

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Unproven Answer (I need to go out soon):

This is the graph:

enter image description here

And the solution is unique, which is:

$10, 14, 15, 20, 21, 28$

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