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Simon's Favorite Factoring Trick

Example and Introduction

Let's say you have the following:
 

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

xy + ax + by = c

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

We start with our equations:

P = 2a + 2b

A = ab

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 * 4, 2 * 2, and 4 * 1 So, we get

(3, 6), (4, 4), (6, 3)

Most Common Cases

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