Data Sufficiency

From Statement 1, 2L + 2W = 28, so L + W = 14. There are a few pairs of positive odd integers that add to 14, say 9 and 5 or 3 and 11, and these will give us different areas, so this is not sufficient.

From Statement 2, the area of an L by L square is 80% bigger than the area of our L by W rectangle. So

L^2 = 1.8 LW
L = 1.8W
L = 9W/5

Since 9W/5 is equal to the integer L, it must be true that W is divisible by 5. It's also an odd integer, but that's really all we know - W could be 5, and L could be 9, or W could be 15 and L could be 27, among other possibilities.

But combining the two statements, W must be 5 and L must be 9, so the answer is C.