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Simon's Favorite Factoring Trick

Example and Introduction

Let's say you have the following:
 

MathJax TeX Test Page $$xy - x - y = 119$$ You can factor the left side, making it $$x(y-1) - y = 119$$ Now, it seems like you're stuck, but by adding 1 on each side, you can factor it. $$x(y-1) - y + 1 = 120$$ $$x(y-1) - (y-1) = 120$$ $$(x-1)(y-1) = 120$$ Now, if the problem is asking for integers, you can just look through the possible factor pairs of 120, and figure out what x and y should be.

General Case and Rule

MathJax TeX Test Page $$xy + ax + by = c$$ You add $ab$ to both sides. $$x(y + a) + by + ab = c + ab$$ $$x(y + a) + b(y + a) = c + ab$$ $$(x + b)(y + a) = c + ab$$ So, when you do those problems, you have to be careful with the signs.
This trick is very useful, and it is commonly asked in AMC competitions.

Another Example

Find the length and width of a rectangle whose area is equal to its perimeter

MathJax TeX Test Page We start with our equations: $$P = 2a + 2b$$ $$A = ab$$ Now, you set them equal. $$2a + 2b = ab$$ $$ab - 2a - 2b = 0$$ Now, we use Simon's Favorite Factoring Trick, and we get $$(a-2)(b-2) = 4$$ So the pairs could be $1 \cdot 4$, $2 \cdot 2$, and $4 \cdot 1$ So, we get $\boxed{(3, 6), (4, 4), (6, 3)}$

Most Common Cases

MathJax TeX Test Page $$xy + x + y + 1 = (x+1)(y+1)$$ $$xy - x - y + 1 = (x-1)(y-1)$$
David Witten

Chinese Remainder Theorem

Square root of a square root