The solution is based on a special and innovative tree that contains all natural numbers. We will call this special tree the Divisor Tree. The root of the tree is vertex number $1$. Each number greater than 1 can be written as a product of prime numbers. The father of any vertex is the number equal to the number divided by the largest prime number dividing it. For example, $3 * 5 * 5 * 7$ has $3 * 5 * 5$ as father, while $2 * 2 * 2$ has $2 * 2$ as father.

\centerline{\includegraphics[scale=.5]{avem.png}}

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In order to understand its structure, you can have a look on the image above. Here you can find a descriptive part of the Divisor Tree. Note that all prime numbers share an edge with root number $1$, and the height of each vertex is equal to the number of times you should divide it by a prime number in order to make the number equal to $1$ (the sum of exponents of the prime numbers in its factorisation).