AlgebraAMC 10

Simon's Favorite Factoring Trick

xy + ax + by + ab = (x+b)(y+a)

Add the constant $ab$ to both sides so the left side factors.

What it says

Simon's Favorite Factoring Trick turns an equation with an $xy$ term and linear terms into a product of two factors equal to a constant. Then integer solutions come from listing factor pairs of that constant. The key move is adding the right constant to both sides.

Show proof

Expand the right side: $(x + b)(y + a) = xy + ax + by + ab$ . So $xy + ax + by = k$ becomes $(x + b)(y + a) = k + ab$ .

When to use it

An equation has an $xy$ term plus linear terms and asks for integer solutions.

Example

How many ordered pairs of positive integers $(x, y)$ satisfy $xy + 2x + 3y = 30$ ?

Show solution

$(x+3)(y+2) = 36$ with $x+3 \ge 4$ and $y+2 \ge 3$ : the factor pairs $(4,9),(6,6),(9,4),(12,3)$ give $4$ solutions. Answer: $4$