A rectangle with positive integer side lengths in has area and perimeter . Which of the following numbers cannot equal ?
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Let the positive integer side lengths of the rectangle be x , y . Then A = x y and P = 2 x + 2 y , and so
A + P = x y + 2 x + 2 y = ( x + 2 ) ( y + 2 ) − 4 ⟹ A + P + 4 = ( x + 2 ) ( y + 2 ) .
So since x , y > 0 , we must be able to represent A + P + 4 as a product m × n with integers m , n > 2 . Now as examples for the given options we have that
1 0 8 + 4 = 1 1 2 = 8 × 1 4 , 1 0 4 + 4 = 1 0 8 = 9 × 1 2 , 1 0 0 + 4 = 1 0 4 = 8 × 1 3 and 1 0 6 + 4 = 1 0 × 1 1 , and so it is possible for A + P to be any of these options.
However, the only possibility for 1 0 2 + 4 = 1 0 6 is 2 × 5 3 , as both 2 and 5 3 are prime, which does not satisfy the requirement that m , n > 2 . Thus A + P cannot equal 1 0 2 .