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
.