Question 77
A *complete number* is a 9 digit number that contains each of the digits 1 to 9 exactly once. The *difference number* of a number $N$ is the number you get by taking the differences of consecutive digits in $N$ and then stringing these digits together. For instance, the difference number of 25143 is equal to 3431. The complete number 124356879 has the additional property that its difference number, 12121212, consists of digits alternating between 1 and 2.
Determine all $a$ with $3 \le a \le 9$ for which there exists a complete number $N$ with the additional property that the digits of its difference number alternate between 1 and $a$.
Determine all $a$ with $3 \le a \le 9$ for which there exists a complete number $N$ with the additional property that the digits of its difference number alternate between 1 and $a$.