# Prime Numbers - Prime Factorization

Prime number is a whole number greater than 1, whose only two whole-number factors are 1 and itself.

$$\mathrm{\mathbb{P}}=\{2,3,5,7,11,13,17,19,23,29\dots \}$$

## Prime Factorization

Any positive integer **m** can be written as a unique product of prime numbers:

$\begin{array}{c}m={{p}_{1}}^{{\alpha}_{1}}\xb7{{p}_{2}}^{{\alpha}_{2}}\xb7...\xb7{{p}_{k}}^{{\alpha}_{k}}\\ {p}_{1,}{p}_{2,}...,{p}_{k}\in \mathrm{\mathbb{P}}\\ {\alpha}_{1},{\alpha}_{k},...,{\alpha}_{k}\in \mathrm{\mathbb{N}}\end{array}$

## Example:

$$\begin{array}{ccc}60& |& 2\\ 30& |& 5\\ 6& |& 2\\ 3& |& 3\\ 1& |& \end{array}$$

$$60={2}^{2}\xb73\xb75$$

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