Question 91
Let $n$ be a positive integer. Prove that the numbers
$$
1^1, 3^3, 5^5, \dots, (2^n - 1)^{2^n - 1}
$$
are in different residue classes modulo $2^n$.
$$
1^1, 3^3, 5^5, \dots, (2^n - 1)^{2^n - 1}
$$
are in different residue classes modulo $2^n$.