The perimeter of a rectangle in units equals its area in units squared. (4,4), (3,6), and (6,3) are three possible pairs of lengths for this rectangle.

Give another pair of positive integral sides for this rectangle or prove why there isn't another pair.

If rectangle has sides (a,b) then  2a + 2b = ab
a = 2b/(b-2)   and   b = 2a/(a-2)
Neither a nor b can be 2 due to a zero denominator;
and neither can be less than 2 or the other would be negative.
(a,b) >= 3
For a to be an integer, the largest b would be if b/(b-2) = 1.5 --> b(max) = 6
(a,b) <= 6
(a,b) can only be from {3,4,5,6}
Neither can be 5 because 2*5/(5-2) = 10/3, not an integer

(a,b) can only be from {3,4,6}

x   2x/(x-2)
3     6
4     4
6     3

Posted by Larry on 2020-10-31 13:07:29