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Answer: 8005
Complete solution:
(a) If 6 is a digit then the product of the digits has 6 as a factor. So (6) is false. But then 6 cannot be a digit. Contradiction. So 6 is not a digit, but the product of the digits is divisible by 6.
(b) If 1 is a digit then by (1) the product of the digits is odd. This means all of the digits are odd. Now the difference between 2 odd numbers is never 1. So (8) is true, and then 8 is a digit. But 8 is even. This contradiction means 1 is not a digit, and the product of the digits is even.
(c) If 4 is a digit then by (4) only 0,2,3,4 can be digits. (Not 1 by (b).) Since 7 is not a digit the number is odd (by (7)) so it must end in an odd digit which can only be 3. This means all of the digits are different by (3) so there are at most 4 digits. This means (5) is true so 5 is a digit. This contradicts (4) so 4 is not a digit, and at least one digit is greater than 4,
(d) If 7 is a digit then (8) cannot be a digit since 8-7=1 contradicts (8). But now since 8 is not a digit there must be 2 digits which differ by 1. The only digits the number can have are: 0,2,3,5,7,9. So it must have 2 and 3 as digits. So by (2) 0 cannot be a digit since it must follow some greater digit. Also by (3) all digits are different. So the number can only have at most 5 digits (2,3,5,7,9). Now by (2), 2 must be the first digit, so the last digit is either 7 or 9, both of which are odd. This means the number is odd which contradicts (7). So 7 is not a digit, and the number is odd.
(e) Recap: So far the number can only have digits 0,2,3,5,8,9 and must be odd, with an even product of digits which is divisible by 6. So it must have at least one odd digit and at least one even digit. Also at least one digit must be greater then 4. Note that 8 and 9 cannot both be digits by (8); also (2) implies (3).
(f) Now if 5 is NOT a digit, then the number has 6 or more digits from {0,2,3,8,9}. If 3 is a digit then all are different, but there are only 5 different possible digits. So no 3. If 2 is a digit then by (2) so is 3. So no 2. Since the number must have an odd digit it must be 9. This means 8 is not a digit (by (8)), and so there must be 2 digits that differ by 1. But only 0 and 9 are left. This contradiction means that 5 must be a digit, and the number has less than 6 digits.
(g) If 8 is NOT a digit then at least 2 digits must differ by 1 and since the digits come from {0,2,3,5,9} they must be 2 and 3. Now if 0 is a digit it has to follow some greater digit which contradicts (2). So no 0. Also since 2, 3, and 5 are all digits, and 2+3=5, then 9 is a digit by (9). By (3) all digits are different and by (2) they are in ascending order, so the number must be 2359. But the sum of these digits is 19 which is prime, so (0) is true, which contradicts the fact that there is no 0. So 8 is a digit.
(h) If 9 is a digit then since 8 is a digit this contradicts (8). So 9 is not a digit.
(i) If 2 is a digit then by (2) 8 is the last digit (since no 9). But this means the number is even, which contradicts (7). So no 2.
(j) If 3 is a digit then, (since 5 and 8 are digits) 3+5=8 which contradicts (9), since 9 is not a digit. So no 3.
(k) Recap: 1,2,3,4,6,7,9 are NOT digits. 5,8 ARE digits. 0 still undecided.
(l) If 0 is NOT a digit then digital product has only 5s and 8s as factors, which means it cannot be divisible by 6. This contradicts (6). So 0 is a digit.
(m) Now, since there is no 3 then by (3) there must be at least 2 equal digits. The digits can only be 0,5,8 all of which are present. If the are two 5s or two 8s then, since 0+5=5 and 0+8=8, this contradicts (9). This means there are 2 or more 0s. If there are 3 zeros then, since 0+0=0, this contradicts (9). So there are exactly two 0s, one 5, and one 8. Since the number is odd by (7) it must end with 5, and since it can't start with 0, it must be 8005. (Note that the sum of the digits is 13, a prime, which agrees with (0).)
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