If the units are integers, what dimensions have equal areas and perimeters?

Question

The units of area and perimeter must be equal

Answer 1

2W + 2L = LW

2L = LW - 2W = W(L - 2)

2L/(L - 2) = W

 

so, L>0 because it I s a physical dimension and cannot be negative

L > 2 because otherwise W<0

L = 3, W = 6

L = 4, W = 4

L = 6, W = 3

 

are there other combinations?

When is 2L/(L - 2) a positive integer?  (positive because W>0)

For numbers greater than 6, which we will call 6+a

2(6+ a) =W[(6+a)-2]

12 + 2a = W (4+a)

(12+2a)/(4+a) = W

This number is just under 3/1 for small values of a

and just over 2/1 for very large values of a.  It can never achieve a integer value.