S.R.
Okay. Based on the fact that the prime factorization consists of a single prime number used multiple times, you already have eliminated it down to either 2 to the 6th or 3 to the 4th, being either 64 or 81. Neither one of those is able to create 5 distinct rectangles, though. 64 can make 4, 81 only 3. So unless I'm missing something (always possible), this doesn't check out.... Let me do some looking.
ETA: Okay, here is the ONLY way I can get this to work. Using 3^4 with a mystery number of 81, you (by my view) only get 3 distinct rectangles. UNLESS, they are counting rectangles differently. IF they consider order making them unique, then you could have 5.
1x81
81x1
3x27
27x3
9x9
That is the ONLY way I can find to get 5 "different" rectangles, although to me, there are only 3. There's NO way to get 5 out of 64, so 81 seems like the best bet to me, but it's certainly not a good question.