When we described the checksum system for binary messages in class, we added the number of $1\text{'}$s in the ASCII representation of the message and tacked on the binary representation of the sum to the end. $$In our example, the checksum was encoded by $8$ bits. $$For this problem, let's suppose that we are going to represent the checksum with $9$ bits instead.

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Of course, representing the sum of a long message might require more than $9$ bits. $$As such, we won't tack on the total sum of all of the $1\text{'}$s$,$ but instead the sum $($mod N$).$What is the most sensible choice for N$?$