If $r\in\Bbb Z\_{\geq0}$ and $m$ is odd then let $2^\ell\mid\binom{m}{2^r}$ and $2^{\ell+1}\nmid\binom{m}{2^r}$. 



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> Is there a way to find if $\ell$ is even or odd without computing $\binom{m}{2^r}$ (assume $\ell$ odd if $m<2^r$)?
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[Lucas theorem](https://en.wikipedia.org/wiki/Lucas%27s_theorem) states $\ell$ is number of carries when $m-2^r$ is added to $2^r$.


However we look for less information and may be we can get away from explicit adding and using other type of operations.



Divisibility criterion of binomial coefficients
The exponent $\ell$ is the same as the maximal $\ell$ for which $2^\ell$ divides $\lfloor m/2^r\rfloor$. This follows from the criterion with carries or may be proved directly. 


