Simon's Favorite Factoring Trick

Applications

This factorization frequently shows up on contest problems, especially those heavy on algebraic manipulation. Usually $x$ and $y$ are variables and $j,k$ are known constants. Sometimes, you have to notice 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. It can be used to solve more than algebra problems, sometimes going into other topics such as number theory.

Fun Practice Problems

Introductory

Two different prime numbers between $4$ and $18$ are chosen. When their sum is subtracted from their product, which of the following numbers could be obtained?

$\mathrm{(A) \ 22 } \qquad \mathrm{(B) \ 60 } \qquad \mathrm{(C) \ 119 } \qquad \mathrm{(D) \ 180 } \qquad \mathrm{(E) \ 231 }$

(Source)

Intermediate

Problem 1

If $kn+54k+2n+108$ has a remainder of $3$ when divided by $5$ , and $k$ has a remainder of $1$ when divided by $5$ , find the value of the remainder when $n$ is divided by $5$ .

$\mathrm{(A) \ 1 } \qquad \mathrm{(B) \ 0 } \qquad \mathrm{(C) \ 4 } \qquad \mathrm{(D) \ 2 } \qquad \mathrm{(E) \ 3 }$

- icecreamrolls8

Solution

We have solution $\boxed{(D)}$ . Note that $kn+54k+2n+108$ can be factored into $(k+2)(n+54)$ using Simon's Favorite Factoring Trick. Now, look at n. Then, since the problem tells us that $k$ has a remainder of $1$ when divided by 5, we see that the $(k+2)$ factor in the $(k+2)(n+54)$ expression has a remainder of $3$ when divided by 5. Now, the $(n+54)$ must have a remainder of $1$ when divided by $5$ as well [3(mod5) x 1(mod5) = 3(mod5)] matching the main expression which has a remainder of $3$ when divided by $5$ . Therefore, since 54 has a remainder of $4$ when divided by $5$ , $n$ must have a remainder of $2$ , so that the entire factor has a remainder of $1$ when divided by $5$ .

- icecreamrolls8

Problem 2

$m, n$ are integers such that $m^2 + 3m^2n^2 = 30n^2 + 517$ . Find $3m^2n^2$ .

(Source)

Solution

$m^2 + 3m^2n^2 = 30n^2 + 517$ $(m^2-10)(3n^2+1)=507$ $507=13*39$ $(3n^2+1)=13,(m^2-10)=39$ $3m^2n^2=588$

Problem 3

(Source) A rectangular floor measures $a$ by $b$ feet, where $a$ and $b$ are positive integers with $b > a$ . 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 $1$ foot around the painted rectangle and occupies half of the area of the entire floor. How many possibilities are there for the ordered pair $(a,b)$ ?