Factorization

A number can be written as a multiplication or product of other numbers. Number $12$ can be written in several ways, based on its different divisors.

$$12=12 \times 1\\12=6\times 2 \\12=4 \times 3$$

And number $13$, being prime, can only be written as this product:$$13=13 \times 1$$ There is a way to write any number as a product of primes.

For example, in the case of number $12$, you first have to find out what prime numbers are its dividers, testing from the smallest to the largest ones to see which of these divisions is exact. For this reason, it is useful to have the list of prime numbers that has been given at the previous level.

To test which primes are divisors of number $12$, for example, we have to build a two-column table where the divisions are held ‘for trial’. In the first section of the left column we write the number, in this case $12$, and in the first box on the right we put the first prime divisor being tested, which will be the smallest one, giving an exact result. We do successive divisions, using the quotient of the first as a dividend of the second, and so on, until arriving at the unit. We will only write in the table the exact divisions. Thus, at the end of the process, the primes numbers that are divisors of $12$ will be in the right column.

In this case, it would be as follows:

$\begin{eqnarray} 12 & | & 2 \\ 6 & | & 2 \\ 3 & | & 3 \\ 1 & & \end{eqnarray}$

The prime divisors of $12$ are $2$ and $3$.$$12=2 \times 2 \times 3$$

This operation is named a Factorisation or Factoring.

Since we can write $2 \times 2$ as a power, it is possible to write the factorisation of $12$ as follows:$$12 = 2^2 \times 3$$

Each number gives a different and unique decomposition. The decomposition in the product of prime numbers is like the ID card or, even better, the DNA, of every number. The prime numbers are like the elementary particles of arithmetics in which any entire number decomposes into.