Equalizing area and perimeter
A nonsquare rectangle has sides of integral length in cm. Its area in square cm is equal to its perimeter in cm. What is the perimeter in cm?
The answer is 18.
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1 solution
Moderator note:
Good approach. As Brian pointed out, SFFT is a more direct way of solving this diophantine equation.
Slight variation: write 2 a + 2 b = a b as a b − 2 a − 2 b = 0 ⟹ ( a − 2 ) ( b − 2 ) = 4 .
(As the rectangle is non-square, assume without loss of generality that a < b .)
Since a , b ≥ 1 and a = b we must actually have a , b > 2 , for otherwise ( a − 2 ) ( b − 2 ) would be non-positive. So since 4 can only be factored as a product of two different positive integers (without concern for order) in one way, namely 1 ∗ 4 , we must have
a − 2 = 1 , b − 2 = 4 ⟹ ( a , b ) = ( 3 , 6 ) ,
resulting in a perimeter of 2 ∗ ( 3 + 6 ) = 1 8 cm.
Let the side lengths be a and b. Perimeter = 2a + 2b, area = ab.
2a + 2b = ab. ab is therefore even. Let a be even. If a = 2, there is no solution. If a = 4, b = 4 and the rectangle would be a square. If a = 6, b = 3. If a > 6, 2 < b < 3, but b is integral.
Therefore a = 6, b = 3, perimeter = area = 18.