There is a Multiple of Every Integer With Digits Consisting Only of Zero and Ones
Suppose
\[n=2^r5^sm, \; r, \; s \ge 0, \; gcd(n,10)=1\]
. There exists a repunit (a number all of whose digits are 1's) equal to some multiple of \[m\]
, say \[km\]
- Proof that p Divides the Repunit R_p If we can find some further multiple that equalises the powers of 2 and 5, so that \[km\]
is multiplied by some power of 10, the resulting number will be the repunit \[kn\]
with a number of 0's appended.Let
\[t= max \{ r, s\} . \]
Then \[(2^{t-r} 5^{t-s})kn=10^tkm\]
is a suitable multiple. 
