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Let the positive integer $a_i (i \in \mathbb N)$ is composed only of digits $1$ and $0$. For example $a_i=101$, but $201 \not=a_i$.

Find the smallest positive integer the $k$, such that every positive integer n can be written as $$n=a_1\pm a_2 \pm a_3 \pm .... \pm a_k$$

My work so far:

If $n=9=10-1$ then $k=2$

If $n=19=10+10-1$ then $k=3$

If $n=48 = 100-11-11-10-10-10$ then $k=6$

If $n=482=100 + 100 + 100 + 101 + 101 - 10 - 10$ then $k=7$

I proved that $k\le9$.

I think the answer $k=7$. But I do not know how to prove it.

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  • $\begingroup$ @MichaelHoefnagel: $$4846=10000-5136=10000-(1111+1011+1011+1001+1001+1)=$$ $$=10000-1111-1011-1011-1001-1001-1$$ $\endgroup$ – Roman83 Jan 1 '17 at 18:10
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    $\begingroup$ Ok great thanks $\endgroup$ – Mike Jan 1 '17 at 18:14

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