2015 AMC 10A Problems/Problem 20
Problem
A rectangle has area and perimeter , where and are positive integers. Which of the following numbers cannot equal ?
Solution
Let the rectangle's length and width be and . Its area is and the perimeter is .
Then . Factoring, this is .
Looking at the answer choices, only cannot be written this way, because then either or would be .
So the answer is .
Also, when adding 4 to 102, you get 106, which has less factors than 104, 108, 110, and 112.
No Solution
I am afraid the problem has an error: the actual sides of the rectangle had to be integers. As stated, every answer choice would work, with one of the sides being , and the other, a half-integer. E.g., for , the sides of the rectangle would be and .