The Million Dollar (Math) Question

Alright boys and girls, here is a math puzzle that may be a wee bit difficult to solve... but if you can do it, you may be freaking genius. So do get your pencils and papers ready!

Don't worry, there is no need for trigonometry or calculus. Its just pure addition and subtraction and the numbers do not exceed 200.

LOL

Here goes:


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15 pigeons can be distributed into 5 coconut trees such that when a they keep 'dancing' by hopping from tree to tree, assuming that they do not repeat their dance twice per turn at any time, they will eventually return to their original distribution.

The dynamics of the 'dance' are as follows:

1 pigeon hops over from the first coconut tree to the second coconut tree

Then,

2 pigeons hop over from the second coconut tree to the third coconut tree,
3 pigeons hop over from the third coconut tree to the fourth coconut tree,
4 pigeons hop over from the fourth coconut tree to the fifth coconut tree,

and finally,

5 pigeons hop from the fifth coconut tree back to the first coconut tree.

After all the hopping is done, the number of pigeons on each coconut tree is tabulated, and a 'dance' cycle is considered complete. It is noted that for a certain configuration of distribution of pigeons, the number of pigeons on each coconut tree can eventually return back to their original number after a certain number of 'dance' cycles.

This configuration is called a 1,2,3,4,5 hopping configuration.

Now, what is the distribution of pigeons if there are 44 pigeons hopping at a 3,7,8,10,16 configuration; if we want it to be such that the pigeons will return to their original distribution after a number of 'dance' cycles?

ie:

3 pigeons hop over from the first coconut tree to the second coconut tree,
7 pigeons hop over from the second coconut tree to the third coconut tree,
8 pigeons hop over from the third coconut tree to the fourth coconut tree,
10 pigeons hop over from the fourth coconut tree to the fifth coconut tree,
16 pigeons hop over from the fifth coconut tree back to the first coconut tree,


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Don't worry, there is a definite answer, and it isn't a complex number.