# Multiples of $5$ in base $2$

As a follow-up question, what can be said about multiples of $5$ in base 2?

I think I must look at the last two digits, but I am not sure what the whole idea might be.

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A starting point for a divisibility test (by 5) in binary would be the observation that $2^4\equiv1\pmod5$. Therefore also $2^{4n}\equiv1\pmod5$ for all natural numbers $n$. From that point on it is quite similar to the familiar "casting nines" method of testing divisibility by nine in base 10.
For example: $$110101001000_2=1101_2\cdot(16)^2+0100_2\cdot16^1+1000\equiv 1101_2+0100_2+1000_2=13+4+8=25\equiv0\pmod5.$$
we can further utilize $2^2=4\equiv-1\pmod 5,$ right? –  lab bhattacharjee Feb 24 '13 at 11:50