cute puzzle

[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]

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

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

[kamil9876]

yep sorry, edited now, forgot the factor of 4 which eventually doubles my answer after square rooting

[zzdfa]

nice
well, here's an easier way:

note that:






you pretty much had it in the first 2 lines  

[kamil9876]

yeah i guess i just included a lot of words and explanations, giving an impression that it had many computations but it didnt. Still i reckon yours has a less computations so it's probably better, but it's harder to see everything fit in like that imo. Nice stuff.