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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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Some Thoughts re(3): Infinite examples, no proof yet. | Comment 13 of 18 |
(In reply to re(2): Infinite examples, no proof yet. by Jer)

Maybe I don't understand your question, Jer, but it is easy to prove that abcd must be a multiple of 2.  

Assume that the LCM is odd.  
Then a, b, c and d are all odd.  
Then a + b + c + d is even.  But a + b + c + d is the LCM, so this is a contradiction.   
Therefore, the initial assumption is wrong, and the LCM must be even.

  Posted by Steve Herman on 2013-04-29 17:03:36
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