Y by narutomath96, Amir Hossein, Adventure10, Mango247
Let
be a positive integer. Show that the numbers
![\[ \binom{2^n - 1}{0},\; \binom{2^n - 1}{1},\; \binom{2^n - 1}{2},\; \ldots,\; \binom{2^n - 1}{2^{n - 1} - 1}\]](//latex.artofproblemsolving.com/2/a/3/2a3629f4a907a9be48ba5faa50dfbf067252d3d6.png)
are congruent modulo
to
,
,
,
,
in some order.
Proposed by Duskan Dukic, Serbia

![\[ \binom{2^n - 1}{0},\; \binom{2^n - 1}{1},\; \binom{2^n - 1}{2},\; \ldots,\; \binom{2^n - 1}{2^{n - 1} - 1}\]](http://latex.artofproblemsolving.com/2/a/3/2a3629f4a907a9be48ba5faa50dfbf067252d3d6.png)
are congruent modulo






Proposed by Duskan Dukic, Serbia