Last winter I found myself locked out of my house because I couldn't remember the 5-digit code.......What was the code ?

Question

Last winter I found myself locked out of my house because I couldn't remember the 5-digit code to open the garage door. I used the following facts to get inside.
1. The second and third digits add up to 9.
2. The first digit is equal to the second digit cubed.
3. The sum of the third and fifth digits is the smallest number with exactly five divisors.
4. The fourth digit is equal to 6 times the second-to-last digit.
5. None of the digits repeat.
What was the code ?

Answer 1

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.

Answer 2

First digit is 8 and second 2 from hints 2 and 5.
So third digit is 7 from hint 1.
The fifth digit is 3 from hint 3.
Fourth is 0.
The code is 82703