# Digit Divisibility

**Number Theory**Level 4

Define a binary string to be a number which is precisely 11 digits long but consists only of 0s and 1s. 01001000100 is such a string (the string does not necessarily have to start with a 1). Also, let the rightmost digit be denoted \( a_{1} \) and label each digit in the obvious fashion up to the first digit, \( a_{11} \). For how many binary strings does \( 3 \mid \displaystyle\sum_{n=1}^{6} a_{2n-1} - \displaystyle\sum_{m=1}^{5} a_{2m} \) ?