mathschallenge.net

Problem

How many rectangles with integral length sides have an area equal in value to the perimeter?

Solution

Let the rectangle measure $x$ by $y$ so that the area = $xy$ and the perimeter = 2x$$ + 2$y$.

	$xy$	=	2$x$ + 2$y$
	$$xy$ 2$y$$	=	2$x$
	$y$($x$ 2)	=	2($x$ 2) + 4
	$y$	=	2 + 4 / ($x$ 2)

As $x$ and $y$ are integer, it is necessary for $x$ 2 to divide into 4.

Therefore $x$ 2 = 1, 2, or 4 $x$ = 3, 4, or 6 and $y$ = 6, 4, or 3 respectively.

Hence there are two unique rectangles with area and perimeter equal: an oblong measuring 6 by 3 and a square with side length 4.

Investigate "equable" triangles: equilateral, isosceles, and scalene.