# Digit Divisibility

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 $$3 \mid \displaystyle\sum_{n=1}^{6} a_{2n-1} - \displaystyle\sum_{m=1}^{5} a_{2m}$$ ?

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