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Kool Numbers (Posted on 2004-11-15) Difficulty: 4 of 5
A collection of positive integers (not necessarily distinct) is called Kool if the sum of all its elements equals their product.

For example, {2, 2, 2, 1, 1} is a Kool set.
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a) Show that there exists a Kool set of n numbers for all n>1

b) Find all Kool sets with sums of 100

c) Find all Kool sets with 100 members.

No Solution Yet Submitted by SilverKnight    
Rating: 3.0000 (5 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Part 3 Done (I think) | Comment 6 of 10 |
(In reply to Part 3 (partial Solution) by Steve Herman)

Assuming the rest are 1's, I think I have an exhaustive search:

{100,2},{34,4},{10,12},{7,4,4},{3,3,3,2,2}

I followed David's attack in showing that we can't have six 3's nor have five 4's. This speeded the Mathematica up, but I would like a witness to this reasoning :-)


  Posted by owl on 2004-11-15 20:13:35
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