Let w is the width and l the length of the rectangle, such that w and l and whole numbers.

The area of the rectangle is Area = w * l satisfying the condition l>w.

Also Perimeter = 2*w + 2*l (perimeter of rectangle).

Since 120 = 1*2*2*2*3*5 we have the following possibilities for width and length:

If w = 1 then l = 120 and Perimeter = 2*1 + 2*120 = 242

If w = 2 then l = 60 and Perimeter = 2*2 + 2*60 = 124

If w = 3 then l = 40 and Perimeter = 2*3 + 2*40 = 86

If w = 4 then l = 30 and Perimeter = 2*4 + 2*30 = 68

If w = 5 then l = 24 and Perimeter = 2*5 + 2*24 = 58

If w = 6 then l = 20 and Perimeter = 2*6 + 2*20 = 52

If w = 8 then l = 15 and Perimeter = 2*8 + 2*15 = 46

If w = 10 then l = 12 and Perimeter = 2*10 + 2*12 = 44

The width w can not be >10 since in that case we will have length l<w (case when w = 12).

This means we have all the possible combinations of width and length written above.

We notice (from the above) that the smallest perimeter (equal to 44) we get for w = 10 cm and l = 12 cm.