Equivalently, the rational number $\frac17=0,\overline{142857}$ can be written as a sum of $n$ real numbers, less than $\frac17$ containing only the digits $0$ and $1$ in their decimal representation. Assume that all the real numbers are positive and that $n<10$. In this case, the least value of $n$ is $8$, for which we have the identity \[\frac17=0,\!\overline1+0,\!\overline{011111}+0,\!\overline{010111}+0,\!\overline{010111}+0,\!\overline{000111}+0,\!\overline{000101}+0,\!\overline{000101}+0,\!\overline{000100}.\] Suppose now that $\frac17$ is the sum of $i>0$ positive real numbers and $j>0$ negative real numbers, with $i+j\leq7$. Denote by $u=0,\!u_1u_2\dots$ (respectively by $-v=-0,\!v_1v_2\dots$) the sum of the positive (respectively negative) real numbers, so that $\frac17+v=u$ (remark that $u$ and $v$ are necessarily strictly less than $1$). Since $i,j<7$, it follows that $v_6\geq3$, so that $j\geq3$ and $i\leq4$. Then, we must have $v_5\geq4$, so that $j\geq4$ and $i\leq3$. In this case we find $u_3\geq3$ and thus $i=3$. This finally implies that $v_2\geq6$, which is impossible, since $j=4$. The least value of $n$ is therefore $8$.