Natural Numbers as Vectors via Factorization? Apparently factorization of integers allows one to interpret natural numbers as vectors so that "$\gcd$ and $\mathrm{lcm}$ become component-wise $\min$ and $\max$". Could anybody explain this in more detail and provide an example or two?
 A: I hope the idea is clear from the following example.


*

*$m=51450=2\times3\times5^2\times7^3$, think of $m$ as the vector $(1,1,2,3,0,0,\ldots)$.  

*$n=16500=2^2\times3\times5^3\times11$, think of $n$ as the vector $(2,1,3,0,1,0,0,\ldots)$.  

*The gcd of $m,n$ is $2\times3\times5^2$, think of this as the vector $(1,1,2,0,0,\ldots)$ in which each element is the minimum of the two corresponding elements above.  

*And similarly for the lcm.

A: Yes, the fact that natural numbers have a unique factorization into prime powers sounds a lot like the theorem that vectors in a vector space have a unique representation (as a linear combination) in terms of the basis vectors (for a given basis).
If we think of the prime numbers are (infinitely many) "basis vectors", we can represent all natural numbers as their "linear combinations". For example
$150 = 2\times 3\times 5^2 = \vec{2}+\vec{3}+2\cdot\vec{5}$
where, of course, the vector "addition" is just multiplication. Similarly
$5500 = 2^2\times 5^3 \times 11 = 2\cdot\vec{2} + 3\cdot\vec{5} + \vec{11}$
Now what is $\gcd(220, 5500)$? Component-wise $\min\lbrace \vec{2}+\vec{3}+2\cdot\vec{5} + 0\cdot\vec{11}, 2\cdot\vec{2} + 0 \cdot \vec{3} + 3\cdot\vec{5} + \vec{11} \rbrace$ = $\vec{2} + 2\cdot\vec{5} = 50$.
Similarly
$\mathrm{lcm}(220, 5500) = \max\lbrace \vec{2}+\vec{3}+2\cdot\vec{5} + 0\cdot\vec{11}, 2\cdot\vec{2} + 0 \cdot \vec{3} + 3\cdot\vec{5} + \vec{11} \rbrace$
$=2\cdot \vec{2} + \vec{3} + 3\cdot \vec{5} + \vec{11} = 16500$.
The zero "vector" would be $1 = 2^0 \times 3^0 \times 5^0 \times \ldots = 0\cdot\vec{2} + 0 \cdot\vec{3} + 0\cdot\vec{5} + \ldots$, and if we allow negative "scalars", then we would be able to express all rational numbers in the same manner. For example, $\frac{22}{7} = \vec{2} - \vec{7} + \vec{11}$.
This is still not a vector space, though, as the scalars only form a ring (the ring of integers), and not a field. This is a module.
