mathematics - Crack the Code #1

You are a bank robber, and you come across a safe. It has a combination lock of 4 digits. Careful research (and stealing papers at the bank) has given you multiple clues as to the code - just barely enough to figure it out.

_{Digits are referred to as A-B-C-D in the clues. "A + B" is the sum of the first and second digit. All math follows the standard order of operations}

Clues

• The sum of the digits is greater than (A * B) - (C * D).


• The product of the digits plus their sum is divisible by A and C only.


• The number is prime


• The first digit is greater than 6

What four digit number matches these criteria? Also, if you want, post your methodology for finding the correct answer, as this will help me in the future.

_{Note: I am pretty sure that only one number matches all these clues. However, I may have miscalculated. Please correct me in the comments.}

Answer

edited to match intended question:

given that the number is prime, the only potentially valid combination is 8017.

8017:

• 8+0+1+7 = 16, 8*0 - 1*7 = -7, 16 > -7


• 8*0*1*7 + 8+0+1+7 = 16, divisible by 8 and 1, but not 7. 0 is undefined.


• 8017 is prime.


• 8 > 6.

This depends on the assumption that 0 does not divide 16 evenly, which depends on your definition. This can go either way - 16/0 is undefined, so you could say that (0 divides 16) is undefined. Or, since there's no m such that m*0 = 16, you could say that (0 divides 16) is false.

Methodology is pretty simple. for all numbers 0000-9999, have a computer check if it matches all conditions.