smc

Define a super-special number to be a number whose decimal expansion only consists of $0$'s and $1$'s. The problem is equivalent to finding the number of super-special numbers necessary to add up to $\frac{1}{7}=0.142857142857\text{hdots}$. This can be done in $8$ numbers if we take $$0.111111\text{hdots},0.011111\text{hdots},0.010111\text{hdots},0.010111\text{hdots},0.000111\text{hdots},0.000101\text{hdots},0.000101\text{hdots},0.000100\text{hdots}$$ Now assume for sake of contradiction that we can do this with strictly less than $8$ super-special numbers (in particular, less than $10$.) Then the result of the addition won't have any carry over, so each digit is simply the number of super-special numbers which had a $1$ in that place. This means that in order to obtain the $8$ in $0.1428\text{hdots}$, there must be $8$ super-special numbers, so the answer is $\boxed{8}$.