An artist has a strange idea that a canvas looks good only if its perimeter is equal to its area. Now the question is how many such rectangles are there with integer sides - rectangles with their perimeter equal to their area? And why?
Solution :
Area = l * w
Perimeter = 2l+2w
area = perimter => l*w = 2l+2w Moving 2l+2w to left hand side, we get 2l+2w+l*w = 0
Adding 4 to both sides we get=> -2l-2w + lw+4 = 4
=> (l-2)(w-2) = 4
The integer pairs with 4 as product are (1,4),(2,2),(4-1)
If l-2 = 1 and w-2=4, we get l = 3 and w = 6
If l-2 = 2 and w-2 = 2, we get l = 4 and w = 4
If l-2 = 4 and w-2 = 1, we again get l = 6 and w = 4
Solutions 1 and 3 are identical.
So the only two rectangles with integers sides and area and perimeter being equal are rectangles with sides 3 and 6 and 4 and 4.
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