The Factorization Method Basic

Video tutorial

Lecture Notes

The factorization method is the method of choice in solving elementary level indeterminate equations. This technique utilizes the fact that the number of divisors of an integer is limited. Therefore, if an equation can be written in the following one $$(\cdots)(\cdots)\cdots(\cdots)=N$$

where $N$ is an integer, and every bracket is a function of required variables ($x$, $y$, $\cdots$, etc), then each bracket must be one of the $N$'s divisors. This means that the original equation can be turned into a system of equations with each bracket equals one of these divisors. (Of course, permutating these divisors may lead to different solutions.)

Note that if the given equation only contains two variables and is reasonably simple, it may be possible to solve one with respect to the other and go from there. Such method (i.e. "solving one with respect to the other") will appear later in other methods such as The Quadratic Method.

Examples

 (2483) How many integer solutions does the equation $(x+1)(y+1)=25$ have?

Comments

Pay extra attention to whether negative divisors are permitted.