There really is no such number......here's why.....
If the number consisted of only one prime factor....it would have to be either a square, a cube, or 3^4 = 81....
But.....the only cube that exists between 50 and 100 is 4^3 = 64 = 2^6
And the divisors of 64 are 1 | 2 | 4 | 8 | 16 | 32 | 64 ......however, we can only form 4 possible rectangles with these divisors - 1 x 64, 2 x 32, 4 x 16 or 8x 8
So......the number must be a square......but the only squares between 50 and 100 are 64 and 81 and....we have seen that 64 isn't possible
And the divsors of 81 are 1 | 3 | 9 | 27 | 81 .........and only 3 rectangles are possible......1 x 81, 3 x 27 and 9 x 9...... [note, for the same reason, 3^4 = 81 isn't possible, either ]
80 would provdie us with 5 rectangles because its dvisors are 1 | 2 | 4 | 5 | 8 | 10 | 16 | 20 | 40 | 80
And the rectangles are : 1 x 80, 2 x 40, 4, 20, 5 x 16 and 8 x 10
But 80 has two prime factors [ 2 and 5] because 80 = 2^4 x 5
So.....no number exactly matches the given criteria......