Question 23
Let $n$ be a positive integer. Show that the numbers
$$
\binom{2^{n}-1}{0}, \quad\binom{2^{n}-1}{1}, \quad\binom{2^{n}-1}{2}, \quad \ldots, \quad\binom{2^{n}-1}{2^{n-1}-1}
$$
are congruent modulo $2^{n}$ to $1,3,5, \ldots, 2^{n}-1$ in some order.
$$
\binom{2^{n}-1}{0}, \quad\binom{2^{n}-1}{1}, \quad\binom{2^{n}-1}{2}, \quad \ldots, \quad\binom{2^{n}-1}{2^{n-1}-1}
$$
are congruent modulo $2^{n}$ to $1,3,5, \ldots, 2^{n}-1$ in some order.