In this blog post, we describe a factorization method that possibly has not been discussed before. It is best to show via an example:

Factor ${\displaystyle 152}$:

We first start by putting ${\displaystyle 152}$ here:

		${\displaystyle 152}$

We then place a ${\displaystyle 0}$ to the left of ${\displaystyle 152}$.

	${\displaystyle 0}$	${\displaystyle 152}$

We define an operation as adding ${\displaystyle 1}$ from the left and removing ${\displaystyle 1}$ from the right. So after one operation, we have:

	${\displaystyle 1}$	${\displaystyle 151}$

After every operation, we check for coprimality (defined as if the two numbers are coprime). If they are coprime, we continue until they are no longer coprime. Here, another operation takes us here.

	${\displaystyle 2}$	${\displaystyle 150}$

Here, the GCF is 2, which implies a common factor. We place 2 on a list and divide the two numbers by 2:

Prime factors ${\displaystyle 2}$	${\displaystyle 1}$	${\displaystyle 75}$

We continue:

Prime factors ${\displaystyle 2}$	${\displaystyle 1}$	${\displaystyle 75}$
Prime factors ${\displaystyle 2}$	${\displaystyle 2}$	${\displaystyle 74}$
Prime factors ${\displaystyle 2}$ ${\displaystyle 2}$	${\displaystyle 1}$	${\displaystyle 37}$
Prime factors ${\displaystyle 2}$ ${\displaystyle 2}$	${\displaystyle 2}$	${\displaystyle 36}$
Prime factors ${\displaystyle 2}$ ${\displaystyle 2}$ ${\displaystyle 2}$	${\displaystyle 1}$	${\displaystyle 18}$

We keep on going, and find out that even at 18 operations, the numbers are still coprime:

Prime factors ${\displaystyle 2}$ ${\displaystyle 2}$ ${\displaystyle 2}$ ${\displaystyle 19}$	${\displaystyle 19}$	${\displaystyle 0}$

To this, we say that the left number is a prime factor of ${\displaystyle 152}$, and we end by removing the temporary storage we used to store the two numbers.

From here, we get the prime factorization of ${\displaystyle 152}$: ${\displaystyle 152=2^{3}\times19}$.

This factorization method is truly unique.