Simon's Favorite Factoring Trick
The General Statement
Simon's Favorite Factoring Trick (SFFT) (made by aops user Complex Zeta https://artofproblemsolving.com/community/user/1233) is often used in a Diophantine equation where factoring is needed. The most common form it appears is when there is a constant on one side of the equation and a product of variables with each of those variables in a linear term on the other side. An extortive example would be: where is the constant term, is the product of the variables, and are the variables in linear terms.
Let's put it in general terms. We have an equation , where , , and are integer constants, and the coefficient of xy must be 1(If it is not 1, then divide the coefficient off of the equation.). According to Simon's Favorite Factoring Trick, this equation can be transformed into: Using the previous example, is the same as:
If this is confusing or you would like to know the thought process behind SFFT, see this eight-minute video by Richard Rusczyk from AoPS: https://www.youtube.com/watch?v=0nN3H7w2LnI. For the thought process, start from https://youtu.be/0nN3H7w2LnI?t=366.
Here is another way to look at it.
Consider the equation .Let's start to factor the first group out: .
How do we group the last term so we can factor by grouping? Notice that we can add to both sides. This yields . Now, we can factor as .
This is important because this keeps showing up in number theory problems. Let's look at this problem below:
Determine all possible ordered pairs of positive integers that are solutions to the equation . (2021 CEMC Galois #4b)
Let's remove the denominators: . Then . Take out the : (notice how I artificially grouped up the terms by adding ).
Now, (you can just do SFFT directly, but I am guiding you through the thinking behind SFFT). Now we use factor pairs to solve this problem.
Look at all factor pairs of 20: . The first factor is for , the second is for . Solving for each of the equations, we have the solutions as .
For more info on the solution: https://www.cemc.uwaterloo.ca/contests/past_contests/2021/2021GaloisSolution.pdf
This factorization frequently shows up on contest problems, especially those heavy on algebraic manipulation. Usually and are variables and are known constants. Sometimes, you have to realize that the variables are not in the form x and y. Additionally, you almost always have to subtract or add the x, y, and xy terms to one side so you can isolate the constant and make the equation factorable.
Fun Practice Problems
- Two different prime numbers between and are chosen. When their sum is subtracted from their product, which of the following numbers could be obtained?
- If has a remainder of when divided by , and has a remainder of when divided by , find the value of the remainder of when is divided by .
We have solution . Note that can be factored into using Simon's Favorite Factoring Trick. Now, look at n. Then, since the problem tells us that has a remainder of when divided by 5, we see that the factor in the expression has a remainder of when divided by 5. Now, the must have a remainder of when divided by as well (because then the main expression has a remainder of when divided by ). Therefore, since 54 has a remainder of when divided by , must have a remainder of , so that the entire factor has a remainder of when divided by .
- are integers such that . Find .
(Source) A rectangular floor measures by feet, where and are positive integers with . An artist paints a rectangle on the floor with the sides of the rectangle parallel to the sides of the floor. The unpainted part of the floor forms a border of width foot around the painted rectangle and occupies half of the area of the entire floor. How many possibilities are there for the ordered pair ?
By Simon's Favorite Factoring Trick:
Since and are the only positive factorings of .
or yielding solutions. Notice that because , the reversed pairs are invalid.
- The integer is positive. There are exactly ordered pairs of positive integers satisfying:
Prove that is a perfect square.
Source: British Mathematical Olympiad Round 2 #1 https://bmos.ukmt.org.uk/home/bmo2-2005.pdf