From point 2, its clear that first digit is either 1 or 8.. (cube of 1 or 2) (else the cube of 3 or more will not be a single digit). But 5th point says none of the digits repeat, so first digit is 8 and second digit is 2.So, as per point 1, third digit is 7. (as second digit is 2).
As per point 4, second-to-last digit is none other that 4th digit itself for a 5 digit number. So its equal to 6 times of itself. Means, its 0.the remaining 5th digit is obtained from point 3. the smallest number with exact five divisors is 12 (2,3,4,6,12). So the fifth digit is (12-3rd digit) = 12-7 = 5.So the number is 82705.