### The 5 Digit Code

Last winter I found myself locked out of my house because I couldn’t remember the 5 digits 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?

The code is “82709”

If we label each digit a, b, c, d and e, we get the following equations.

1. b + c = 9

2. a = b^{3}

3. c + e = 16 (since 16 is the smallest number with five divisors – 1, 2, 4, 8 and 16).

4. d = 6 * d (The fourth and second-to-last digits are the same number, meaning d must be zero to satisfy the equation)

5. b must be zero, one or two (b can’t be three because that makes a=27, which isn’t a single digit). Zero and one result in duplicate digits (00907 and 11808 respectively) so the only remaining value for b is 2, giving us 82709 and warmth.