AMC10 2015 10B, Problem 3

Isaac has written down one integer two times and another integer three times. The sum of the five numbers is $100$, and one of the numbers is $28.$ What is the other number?

Solution: Let the first number be $x$ and the second be $y$. We have $2x+3y=100$. We are given one of the numbers is $28$. If $x$ were to be $28$, $y$ would not be an integer, thus $y=28$.  $2x+3(28)=100$, which gives $x=\fbox{8}$.

Answer: 8