Question 44
Let $n$ be a positive integer. Prove that the numbers
$$
1^{1}, 3^{3}, 5^{5}, \ldots,\left(2^{n}-1\right)^{2^{n}-1}
$$
all give different remainders when divided by $2^{n}$.
$$
1^{1}, 3^{3}, 5^{5}, \ldots,\left(2^{n}-1\right)^{2^{n}-1}
$$
all give different remainders when divided by $2^{n}$.