# Problem of the Week Problem D and Solution Choices

## Problem

Matilda and Yolanda each chose an integer. When Matilda’s integer is multiplied by the sum of Matilda’s and Yolanda’s integers, the product is $$299$$.

If Matilda’s integer is smaller than Yolanda’s integer, determine all possible pairs of integers that Matilda and Yolanda could have chosen.

## Solution

Let $$x$$ represent Matilda’s integer and $$y$$ represent the Yolanda’s integer. We’re given $$x<y$$.

The sum of the two integers is $$x+y$$. We multiply this sum by Matilda’s integer, $$x$$. The resulting expression is $$x(x+y)$$. Thus, we want to find all pairs of integers satisfying $$x(x+y)=299$$ with $$x<y$$.

We want the product of two integers to be $$299$$. The factors of $$299$$ are $$\pm 1$$, $$\pm 13$$, $$\pm 23$$, $$\pm 299$$. Thus, the possible values for $$x$$ are $$\pm 1$$, $$\pm 13$$, $$\pm 23$$, $$\pm 299$$.

In the following table, we list all the possible values for $$x$$ and then determine the corresponding value for $$y$$. If $$x<y$$, then this is a valid possibility. For example, if $$x=1$$, then $$x+y$$ must be $$299$$. Therefore, $$y$$ must be $$298$$. Since $$x<y$$, one possibility is Matilda chooses $$1$$ and Yolanda chooses $$298$$.

$$x$$ $$x+y$$ $$y$$ $$x<y$$?
$$1$$ $$299$$ $$298$$ Yes
$$13$$ $$23$$ $$10$$ No
$$23$$ $$13$$ $$-10$$ No
$$299$$ $$1$$ $$-298$$ No
$$-1$$ $$-299$$ $$-298$$ No
$$-13$$ $$-23$$ $$-10$$ Yes
$$-23$$ $$-13$$ $$10$$ Yes
$$-299$$ $$-1$$ $$298$$ Yes

Therefore, there are four pairs of integers that Matilda and Yolanda could have chosen. Matilda could have chosen $$1$$ and Yolanda chose $$298$$, Matilda could have chosen $$-13$$ and Yolanda chose $$-10$$, Matilda could have chosen $$-23$$ and Yolanda chose $$10$$, or Matilda could have chosen $$-299$$ and Yolanda chose $$298$$.