Difference between revisions of "2015 AMC 10A Problems/Problem 20"
Happycowzh (talk | contribs) (→Solution) |
Drkotlovmath (talk | contribs) (→Solution) |
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Also, when adding 4 to 102, you get 106, which has less factors than 104, 108, 110, and 112. | Also, when adding 4 to 102, you get 106, which has less factors than 104, 108, 110, and 112. | ||
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+ | ==No Solution== | ||
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+ | 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 <math>2</math>, and the other, a half-integer. E.g., for <math>102</math>, the sides of the rectangle would be <math>2</math> and <math>49/2</math>. |
Revision as of 20:07, 4 February 2015
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
.