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.

Let the k non-one numbers of the set be a_{1}, a_{2},..., a_{k}. Let P_{k} be their product, and let S_{k} be their sum. We then demand that f(a_{1},a_{2},...,a_{k}) = 0, where f is a function from [2,infty)^k to (-infty, infty) such that

f(a_{1},a_{2},...,a_{k}) = P_{k} - S_{k} + (k-100).

Now for k>2, it is clear that this function increases with respect to each of the a_{i}, since letting a_{i}->a_{i}+1 inccreases S_{k} by 1 and increases P_{k} by more than 1. But for k>6, then, we have:

f(a_{1},a_{2},...,a_{k}) >= f(2,2,...,2)

= 2^k - 2k + (k-100)

= 2^k - k - 100

> 0.

This implies that the only solutions are with k<7. We can do a brute force search through those possibilities, by running a simple program. I don't have time to do this. Maybe somebody else can do this.