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cute puzzle

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zzdfa:
hi guyz

i thought the solution to this was quite elegant:

I have 4 (not necessarily integer) numbers, say m,n,o,p.

Now obviously there are 6 ways to multiply them:

mn no op
mo np
mp

I tell you that of these 6 products, 5 of them are:

2,3,4,5,6. What's the 6th one equal to?

kamil9876:
I often show public displays of affection towards prime factors on this forum.

TrueTears:

--- Quote from: kamil9876 on July 03, 2009, 12:05:05 am ---I often show public displays of affection towards prime factors on this forum.

--- End quote ---
You and your prime.

kamil9876:
we can assume, WLOG, that is what we do not have. Notice that:


But the only products satisfying such a ratio are 2 and 4;3 and 6. Hence we can assign, WLOG, that and so this leaves us with .






similairly:





Discarding the negative because it yields a contradiction since if op is negative then one of the numbers must be negative. Assume p is negative, this implies that n is negative since np is positive. Now if n is negative then that implies that m is negative since mn is positive. But that implies that mo must be negative(since we assumed o is positive), hence implying that one of the 5 products you provided us with is negative which yields a contradiction.
Same argument can be applied if p is positive hence the law of the excluded middle implies that cannot be negative.

zzdfa:
op cant be 6/5, check your working

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