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Dark Divisibility (Posted on 2013-04-27) Difficulty: 3 of 5
The natural numbers a,b,c,d are such that their least common multiple equals a+b+c+d. Prove that abcd is divisible by 3 or by 5.

No Solution Yet Submitted by Danish Ahmed Khan    
Rating: 5.0000 (2 votes)

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Hmmm. | Comment 1 of 18

Let (say) a=1, b=1,c=2,d=4. Then a+b+c+d=8, and abcd=8, which is not divisible by 3 or 5. I wouldn't be surprised if abcd must be divisible by 2 or 3; for one possible line of attack, compare http://perplexus.info/show.php?pid=7987, http://perplexus.info/show.php?pid=8105;  though, since this was not the question asked, I haven't considered the possibility in any detail.

NOTE: The above is incorrect. As Wikipedia explains 'The LCM of more than two integers is also well-defined: it is the smallest integer that is divisible by each of them.'

Edited on April 28, 2013, 10:05 am
  Posted by broll on 2013-04-27 12:08:44

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